Solution of Simultaneous Linear Equations graphically | ICSE Class 9 Maths Notes

This note covers linear equations in two variables, ordered pairs, plotting straight lines, graphical solutions of simultaneous equations, intersecting lines, parallel lines, coincident lines, verification of solutions and problems expressed through pairs of equations.

What does a solution of simultaneous linear equations mean?

How are the variables and coefficients defined?

A variable is a symbol representing a number whose value can vary or is unknown. Here, x and y denote the two variables. An equation states that the expressions on its two sides are equal.

Definition: A linear equation in two variables can be written as ax + by + c = 0. Here a and b are numerical coefficients, the numbers multiplying x and y; c is the constant term, a fixed number. The coefficients a and b are not both zero.

The standard form is ax + by + c = 0. The letters a, b and c describe its structure; the equations solved here have numerical coefficients. A coefficient can be zero, provided the other variable's coefficient is not also zero.

A solution is a pair of values that makes the equation true. The notation (x, y) is an ordered pair: its first entry gives x and its second gives y. The order matters because each entry belongs to a particular variable.

Why must the same pair satisfy both equations?

Simultaneous equations are equations considered together, with the same values required for their shared variables. A common solution satisfies both equations. Finding a pair that satisfies just one equation does not complete the task.

Worked example 1. For 3x + 2y = 12, check the ordered pairs (2, 3), (4, 0) and (1, 3).

Answer: For (2, 3), substitution gives 3 × 2 + 2 × 3 = 12, so it is a solution. For (4, 0), 3 × 4 + 2 × 0 = 12, so this is also a solution.

For (1, 3), the left side is 3 × 1 + 2 × 3 = 9, which is not 12. This pair is not a solution. Substitution means replacing each variable by its proposed value.

This example distinguishes checking a solution from finding every solution. A single linear equation in two variables has infinitely many solutions. A second equation imposes another condition, so the common solutions must be examined separately.

How do ordered pairs become points and straight lines?

How should the coordinate axes be read?

The Cartesian plane is a plane containing two number lines at right angles. The horizontal line is the x-axis; the vertical line is the y-axis. These are the coordinate axes. Their meeting point is the origin, written O, with coordinates (0, 0).

The coordinates of a point specify its position. Its x-coordinate gives its horizontal position, and its y-coordinate gives its vertical position. Positive values lie rightwards or upwards from the origin; negative values lie leftwards or downwards.

A scale states the numerical value represented by a chosen distance on an axis. Mark equal numerical steps at equal distances on each axis. Keep the same axes and scales for both equations so their points can be compared directly.

The graph of an equation consists of the points representing its solutions. For a linear equation, these points lie on a straight line. Every point on that line satisfies the equation, and every solution gives a point on the line.

How are plotting points calculated?

Choose a value for one variable, substitute it into the equation and calculate the other variable. Record the two values together. A table of values keeps the corresponding values paired correctly before they are plotted.

Worked example 2. Find points for drawing the graph of 2x + 5y = 0.

Answer: At x = 0, the equation gives y = 0. At x = 1, it gives y = −2/5. At x = 5, it gives y = −2. Thus the points are (0, 0), (1, −2/5) and (5, −2).

Plot the points and draw the straight line through them. Two distinct points, meaning two different points, determine a straight line. A third calculated point provides a useful check on the calculation and plotting.

Note: Setting x = 0 and then y = 0 in 2x + 5y = 0 produces the origin both times. This is one point, not two distinct points. Use another value, such as x = 5, to obtain a second point.

How is a pair of equations solved graphically?

What sequence connects calculation to the graph?

The graphical method solves simultaneous equations by drawing the straight line for each equation on the same Cartesian plane. A point shared by both lines represents values of x and y that satisfy both equations.

1. Write the two equations clearly and calculate at least two distinct solution pairs for each equation.
2. Draw and label the coordinate axes, mark the origin, and choose scales that allow the calculated points to be plotted.
3. Plot each equation's points separately, preserving the order and signs of the coordinates.
4. Draw the straight line through each set of points, extending it as needed, and label it with its equation.
5. Read the coordinates of the common point, if there is one, and verify them in both original equations.

Result: Intersecting lines have one common solution

Intersecting lines meet at a point. If the two lines meet at one point, its coordinates give the unique solution, meaning exactly one common solution. A pair with at least one solution is called consistent.

Worked example 3. Solve x + 3y = 6 and 2x − 3y = 12 graphically.

Answer: The first equation gives y = (6 − x)/3. Using x = 0 and x = 6 gives (0, 2) and (6, 0). The second gives y = (2x − 12)/3. Using x = 0 and x = 3 gives (0, −4) and (3, −2).

Draw the two lines through their respective points. They meet at (6, 0), so x = 6 and y = 0. Verification gives 6 + 3 × 0 = 6 and 2 × 6 − 3 × 0 = 12.

The graph labels A(0, 2), B(6, 0), P(0, −4) and Q(3, −2). A, B, P and Q are names of plotted points. The two equation lines meet at B(6, 0).

See Fig. 13.14 in your NCERT textbook

The zero y-coordinate does not mean that no solution exists. It places the common point on the x-axis. The pair is consistent because the same ordered pair satisfies both equations.

What does a graph of parallel lines tell us?

Result: Distinct parallel lines have no common solution

Parallel lines here means distinct straight lines in the same plane that do not meet, however far they are extended. Their graphs share no point. Consequently, there is no ordered pair satisfying both equations, and the pair is called inconsistent.

Each equation still has its own solutions. The absence of a solution refers to the pair considered together. This distinction matters: two correctly drawn lines can demonstrate that simultaneous requirements cannot be met.

Worked example 4. Determine graphically whether x + 2y − 4 = 0 and 2x + 4y − 12 = 0 have a common solution.

Answer: Rewrite the equations as x + 2y = 4 and 2x + 4y = 12. The first line passes through (0, 2) and (4, 0). The second passes through (0, 3) and (6, 0).

Plot both pairs of points and draw the two lines. They are distinct and parallel, so they have no intersection and no common solution. The pair is inconsistent.

One line joins R(0, 2) and S(4, 0); the other joins P(0, 3) and Q(6, 0). R, S, P and Q label points. Both lines descend from left to right and remain separate.

See Fig. 13.15 in your NCERT textbook

How can the conclusion be checked?

Dividing the second equation by 2 gives x + 2y = 6. The first requires that same expression, x + 2y, to equal 4. For the same values of x and y, it cannot equal both 4 and 6.

This calculation supports the graph's conclusion. Merely failing to see an intersection within a small drawing is insufficient: intersecting lines may meet beyond the part drawn. For this pair, the incompatible equalities establish why extending the lines will not produce a solution.

Why do coincident lines give infinitely many solutions?

Result: Coincident lines share every point

Coincident lines occupy the same straight line. Both equations are satisfied by every point on that line. The pair therefore has infinitely many common solutions, even though the drawing shows just one visible line.

Equations with the same solutions are called equivalent equations. A pair of equivalent linear equations is a dependent pair. Such a pair is consistent because common solutions exist; consistency does not require the solution to be unique.

Worked example 5. Examine the graphs of 2x + 3y = 9 and 4x + 6y = 18.

Answer: Multiplying every term of the first equation by 2 gives the second equation. The points (0, 3), (3, 1) and (4.5, 0) lie on the common line. Plot them and label the line with both equations.

The graphs coincide. Every point on their shared line solves both equations, so the pair is dependent and consistent, with infinitely many solutions.

The graph shows one descending straight line labelled with both 2x + 3y = 9 and 4x + 6y = 18. The points (0, 3), (3, 1) and (4.5, 0) are marked on it.

See Fig. 13.16 in your NCERT textbook

How are the three possible outcomes different?

Do not stop after finding one point that satisfies a dependent pair. That point is an example of a solution, but the full conclusion concerns every point on the common line. Listing one ordered pair alone would conceal the infinitely many other solutions.

Likewise, two differently written equations need not give two distinct graphs. In this example, the second equation repeats the first condition after multiplication. The numerical appearance changes, but the set of points satisfying the condition does not.

How can two conditions in a word problem be graphed?

How should the unknown quantities be named?

Start by defining what each variable represents. Keep those meanings unchanged while forming both equations. Translate each separate condition into an equation before calculating points. If a quantity has a unit, include it when defining the variable and interpreting the answer.

For a problem about numbers of activities, x and y represent counts. For a problem about costs, they could represent amounts of money. The meaning of the variables determines what the final coordinates say about the original situation.

Worked example 6. Akhila plays Hoopla, a ring-throwing game, half as many times as she rides the Giant Wheel. Using ₹ for rupees, each ride costs ₹3 and each game costs ₹4. She spends ₹20. Find the number of rides and games graphically.

Answer: Let x be the number of rides and y the number of games. The first condition gives y = x/2. The spending condition gives 3x + 4y = 20.

For y = x/2, use (0, 0) and (4, 2). For 3x + 4y = 20, use (0, 5) and (4, 2). Draw both lines on the same axes. Their intersection is (4, 2), giving four rides and two games.

Check the conditions: two games are half of four rides, and 3 × 4 + 4 × 2 = 20. Both the number condition and the spending condition are satisfied.

How does the answer return to the situation?

The graph yields an ordered pair, but the conclusion should name the quantities. Here x = 4 refers to rides and y = 2 refers to games. Reversing the meanings after drawing the graph would produce an incorrect interpretation.

The two equations also have different roles. The first expresses the relation between the counts. The second expresses total spending. A proposed answer must satisfy both roles. The graphical method locates the point at which those two requirements hold together.

Plotting the origin for the first equation does not claim that Akhila took no rides. It is a point used to draw that equation's line. The point solving the complete problem is the intersection with the spending line.

Can a correct graphical answer contain a zero?

How should an intersection on an axis be interpreted?

A point on the x-axis has y-coordinate zero, and a point on the y-axis has x-coordinate zero. An intersection on an axis can therefore be a valid simultaneous solution. Its meaning depends on the quantities represented by the variables.

A zero value is a specific value of a variable. It is different from no solution, which means that no pair of values satisfies both equations. The graph and substitution together distinguish these cases.

Worked example 7. Champa buys some pants and skirts. The number of skirts is two less than twice the number of pants and four less than four times the number of pants. Find both numbers graphically.

Answer: Let x denote the number of pants and y the number of skirts. The conditions give y = 2x − 2 and y = 4x − 4.

For the first equation, use (2, 2) and (0, −2). For the second, use (0, −4) and (1, 0). Draw the corresponding lines. They intersect at (1, 0), so Champa buys one pair of pants and no skirts.

Verification gives 2 × 1 − 2 = 0 and 4 × 1 − 4 = 0. Both statements give the required number of skirts.

The lines y = 2x − 2 and y = 4x − 4 pass through the plotted points (2, 2), (0, −2), (0, −4) and (1, 0). They meet at (1, 0) on the x-axis.

See Fig. 3.2 in your NCERT textbook

Negative coordinates in the plotting tables help construct the mathematical lines. They do not represent actual purchases of negative numbers of skirts. The common point supplies the answer, which must then be interpreted within the original problem.

How can another pair be solved and checked independently?

How are the two value tables kept separate?

For a fresh pair, repeat the full method rather than copying the appearance of an earlier graph. Different equations can require different plotting points. Keep each calculated point attached to the equation that produced it.

Worked example 8. Solve 2x + y − 6 = 0 and 4x − 2y − 4 = 0 graphically.

Answer: Rewrite the first equation as y = 6 − 2x. Setting x = 0 gives y = 6; setting x = 3 gives y = 0. Its line therefore passes through (0, 6) and (3, 0).

Rewrite the second equation as y = 2x − 2. Setting x = 0 gives y = −2; setting x = 1 gives y = 0. Its line passes through (0, −2) and (1, 0).

Plot these points on common axes and draw both straight lines. They intersect at (2, 2). Thus x = 2 and y = 2 is the unique solution.

Check in the original equations: 2 × 2 + 2 − 6 = 0 and 4 × 2 − 2 × 2 − 4 = 0. Both equalities hold, confirming the common point.

What does verification establish?

Checking in the original equations confirms that no sign error in rearrangement has changed the problem being solved. A point satisfying a wrongly rearranged equation may still fail the original equation.

Here the intersection was not included among the points initially chosen for either line. The point (2, 2) lies between (0, 6) and (3, 0) on the first line, but beyond (1, 0) on the second line. Extend the second line through (0, −2) and (1, 0) to reach the intersection.

The answer includes two linked conclusions. The numerical conclusion is (2, 2). The graphical conclusion is that the lines intersect at one point. Together they state both the solution and why it is unique.

How can graphical solutions be made accurate and complete?

What should be checked before reading the intersection?

Graphical work depends on correct calculations and correct plotting. A well-drawn line through incorrect points represents the wrong equation. A correct table followed by reversed coordinates also gives the wrong graph. Check the transition from equation to table and from table to points.

1. Read each equation again and confirm the sign of every term when rearranging it.
2. Substitute the tabulated pairs into their own equation to check that the calculated values belong together.
3. Check the labels and scales of both axes, including the directions for negative coordinates.
4. Ensure each straight line passes through its calculated points and is labelled with the correct equation.
5. Read the intersection as an ordered pair, then substitute both values into both original equations.

What limitation does the method have?

The graphical method is not convenient when the solution has non-integral coordinates, meaning coordinates that are not integers. Integers are whole numbers, their negatives and zero. There is every possibility of making mistakes while reading such coordinates.

An intersection may lie between marked divisions. An approximate reading is a value estimated from the drawing. Do not treat an uncertain reading as an exact value merely because the lines appear to meet there. Verification tests the proposed pair against the actual equations.

What should the final statement contain?

For intersecting lines, state both coordinates and the corresponding values of the variables. For parallel lines, state that there is no common solution. For coincident lines, state that there are infinitely many common solutions, represented by all points on the shared line.

For a word problem, finish with the quantities asked for, using their original meanings and units. The intersection alone is an intermediate mathematical result until it has been connected to those quantities.

Note: A complete graphical solution links the equations, the calculated points, the drawn lines, the common-point conclusion and the verification. The drawing explains where the answer comes from; substitution checks whether the proposed values satisfy both conditions.

Glossary

• Variable — A symbol standing for a numerical quantity whose value may vary or be unknown.
• Coefficient — The numerical factor multiplying a variable in an algebraic expression or equation.
• Linear equation in two variables — An equation expressible as ax + by + c = 0, with numerical constants a, b and c, and a and b not both zero.
• Ordered pair — Two values written in a fixed order, giving the x-value first and the y-value second.
• Cartesian plane — The plane containing perpendicular coordinate axes used to locate points by ordered pairs.
• Origin — The point where the two coordinate axes meet, with coordinates (0, 0).
• Common solution — An ordered pair that satisfies both equations of a simultaneous pair.
• Graphical method — Solving equations by drawing their graphs and examining their common points.
• Unique solution — The single common ordered pair represented by the intersection of two equation lines.
• Parallel lines — Distinct straight lines in the same plane that do not meet when extended.
• Coincident lines — Lines occupying the same position, with every point shared by both graphs.
• Consistent pair — A pair of equations having at least one common solution.
• Inconsistent pair — A pair of equations for which no common solution exists.
• Dependent pair — A pair of equivalent linear equations with infinitely many common solutions.

Common errors and misconceptions

• Misconception: A point satisfying either equation solves the pair. Correct: The same ordered pair must satisfy both equations. Check both before accepting the proposed solution.
• Misconception: The coordinates can be written in either order. Correct: In (x, y), the first value belongs to x and the second to y. Their roles are fixed.
• Misconception: Two equations must have one common solution. Correct: Their graphs may intersect, remain parallel or coincide, giving one, no or infinitely many common solutions respectively.
• Misconception: Coincident graphs have only one solution because only one line is visible. Correct: Every point on the shared line satisfies both equations, giving infinitely many solutions.
• Misconception: A zero coordinate means no solution. Correct: Zero is a numerical value. For x + 3y = 6 and 2x − 3y = 12, the common solution is (6, 0).
• Misconception: Setting each variable to zero must give two plotting points. Correct: For a line through the origin, both calculations may give (0, 0). Another distinct point is needed.
• Misconception: Lines that do not meet inside the drawn area must be parallel. Correct: Their intersection may lie outside that area. Extend the drawing or check the equations before deciding.

Exam-style questions with model answers

Q1. What is a common solution of two linear equations? What does it represent on their graphs? [2 marks]

1. A common solution is an ordered pair of values that satisfies both equations simultaneously.
2. On the graphs, it represents a point lying on both equation lines.

Q2. Check whether (6, 0) solves both x + 3y = 6 and 2x − 3y = 12. [2 marks]

1. Substituting x = 6 and y = 0 into the first equation gives 6 + 3 × 0 = 6.
2. The second gives 2 × 6 − 3 × 0 = 12. Both are true, so (6, 0) is a common solution.

Q3. Explain the number of common solutions when the graphs of two linear equations intersect at one point, are distinct and parallel, or coincide. Classify each pair as consistent or inconsistent. [3 marks]

1. Intersecting lines share exactly one point. Its coordinates satisfy both equations, giving a unique solution, so the pair is consistent.
2. Distinct parallel lines have no common point. No ordered pair satisfies both equations, so the pair is inconsistent.
3. Coincident lines share every point on one line. They have infinitely many common solutions and form a dependent, consistent pair.

Q4. Explain graphically why x + 2y − 4 = 0 and 2x + 4y − 12 = 0 have no common solution. Give two plotting points for each line. [4 marks]

1. For x + 2y = 4, setting x = 0 and then y = 0 gives the points (0, 2) and (4, 0).
2. For 2x + 4y = 12, the corresponding points are (0, 3) and (6, 0).
3. Plot these pairs on the same coordinate axes and draw the straight lines. They are distinct and parallel.
4. They have no common point, so the pair has no solution. This agrees with the incompatible requirements x + 2y = 4 and x + 2y = 6.

Q5. Solve x + 3y = 6 and 2x − 3y = 12 graphically. Show the plotting points, state the nature of the solution and verify your answer. [5 marks]

1. Rearrange the first equation as y = (6 − x)/3. Choose x = 0 and x = 6 to obtain (0, 2) and (6, 0).
2. Rearrange the second as y = (2x − 12)/3. Choose x = 0 and x = 3 to obtain (0, −4) and (3, −2).
3. Plot all four points on common labelled axes. Draw and label the straight line through each equation's pair of points, extending the lines to their intersection.
4. The lines intersect at (6, 0). Therefore x = 6 and y = 0 is the unique solution, and the pair is consistent.
5. Substitution gives 6 + 3 × 0 = 6 and 2 × 6 − 3 × 0 = 12. Both original equations are satisfied.

Q6. Akhila plays Hoopla half as many times as she rides the Giant Wheel. A ride costs ₹3 and a game costs ₹4. She spends ₹20. Form equations, solve graphically and check the numbers of rides and games. [6 marks]

1. Let x be the number of Giant Wheel rides and y the number of Hoopla games. These meanings apply throughout the solution.
2. The number condition gives y = x/2. The total spending condition gives 3x + 4y = 20.
3. For y = x/2, calculate (0, 0) and (4, 2). Plot these points and draw the first straight line.
4. For 3x + 4y = 20, calculate (0, 5) and (4, 2). Draw the second line on the same axes.
5. The common point is (4, 2). Thus Akhila takes four rides and plays two games.
6. Two is half of four, and 3 × 4 + 4 × 2 = 20. Both the count relation and the total spending are correct.

Q7. For 2x + 3y = 9 and 4x + 6y = 18, explain why the graphs coincide and why giving just one ordered pair is an incomplete description of the solutions. [3 marks]

1. Multiplying every term of 2x + 3y = 9 by 2 gives 4x + 6y = 18. The two equations are equivalent.
2. Their graphs therefore occupy the same straight line. Every point on that line satisfies both equations.
3. There are infinitely many common solutions. One ordered pair gives an example, but does not describe all the solutions of this dependent, consistent pair.

Key takeaways

• A solution of simultaneous equations is one ordered pair whose values satisfy both equations together.
• Calculate at least two distinct points for each line, then plot both lines on common labelled axes.
• Intersecting equation lines give a unique common solution, read from the coordinates of their intersection.
• Distinct parallel lines have no common point, so the corresponding pair of equations is inconsistent.
• Coincident lines give infinitely many common solutions because every point on the shared line satisfies both equations.
• A zero coordinate places a solution on an axis; it does not mean that the pair has no solution.
• Verify the coordinates read from a graph by substituting the same values into both original equations.
• For word problems, define the variables before forming equations and interpret the final coordinates using those definitions.

Test yourself

What does the first entry in an ordered pair (x, y) represent?

It represents the x-coordinate, specifying the point's horizontal position relative to the origin.

Why does a common point of two equation lines solve both equations?

Every point on an equation's line satisfies that equation. A common point lies on both lines, so its coordinates satisfy both.

What are the two plotting points obtained by setting x = 0 and x = 6 in x + 3y = 6?

The corresponding values are y = 2 and y = 0, giving (0, 2) and (6, 0).

What is wrong with treating the origin twice as two points for 2x + 5y = 0?

The same point has been repeated. A straight line requires two distinct points for its position to be determined.

What is the solution of x + 3y = 6 and 2x − 3y = 12?

The lines meet at (6, 0), giving the unique solution x = 6 and y = 0.

Can a consistent pair have infinitely many solutions?

Yes. A dependent pair has coincident graphs and infinitely many common solutions, so it is consistent.

If x is the number of pants and y the number of skirts, what does (1, 0) mean?

It means one pair of pants and no skirts; the zero is a valid value of y.

Why can a graphical reading with non-integral coordinates be difficult?

The intersection may fall between marked divisions, making an exact reading inconvenient and allowing mistakes in the coordinates read.