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Graphically Solving a Pair of Linear Equations
When the mathematical expressions of variables and constants along with the mathematical operations form an equation of the highest degree one, it is said to be a linear equation. A linear equation is an algebraic equation between variables that gives a straight line when plotted on a graph. A linear equation of one variable is of the form ax + b = 0 where x is the variable. Linear equations of two variables are of the form ax + by + c = 0 where x and y are the two variables and c is the constant. A pair of linear equations can be solved and represented using two basic methods: graphical method and algebraic method. In this minilesson, we will explore solving a system of two linear equations using graphical method through solved examples, linear equations worksheets, and interactive questions.
Solving Pair of Linear Equation Graphically
Every linear equation consists of variables. Linear equations are of the first order and they may involve one or two variables. When it comes to solving linear equations using graphical method the basic approach is to represent them as straight lines on a graph and find the points of intersection, if any. We can obtain at least two solutions easily by substituting the values for x, finding the x and y intercepts and plotting them geometrically on the graph. Let us have a look at the standard form of a pair of linear equations here.
a_{1}x + b_{1}y = c_{1} ….(1)
a_{2}x + b_{2}y = c_{2} ….(2)
Solution for the equations varies according to the position of the lines. Let us discuss about the solutions obtained in linear equations in two variables.
Types of Solutions
 Consistent : The pair of equations is said to be consistent, if the two lines are intersecting at the same point, then the point gives unique solution for both the equations.
 Dependent : The pair of equations is said to be dependent, if the two lines coincide, then in this case there are infinitely many solutions. Each and every point on a line becomes a solution.
 Inconsistent : The pair of equations is said to be inconsistent, if the two lines are parallel, then in this case there is no solution.
Look at the table below showing the conditions for the given equations.
a_{1}x + b_{1}y = c_{1} ….(1)
a_{2}x + b_{2}y = c_{2} ….(2)
Here a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are the coefficients of the general equations. In the graphical representation m_{1} and m_{2} represents two lines.
How do you Solve Pair of Linear Equations Graphically?
Solving a linear equation on graph depends on the variables. If it is pair of linear equations in two variables it is represented by two lines and if it is single variable it is represented by a single line. We know that solution for the equations varies according to the position of the lines. Let us look at the following steps to solve a pair of linear equations graphically.
Given equations:
y = x  4........(1)
y = x + 2.......(2)
 Step 1: Observe the equations. They are of the form y = mx + b, where m is the slope.
 Step 2: Find the intercepts of the graph of the given equations.
 Step 3: First, find the xintercept of first equation y = x  4 by substituting x = 0, we get y = 4. For y = 0, x = 4. We get intercepts for equation 1 as (0, 4) and (0, 4).
 Step 4: Find the xintercept of second equation y = x + 2 by substituting x = 0, we get y = 2. For y = 0, x = 2. We get intercepts for equation 2 as (0, 2) and (0, 2).
 Step 5: Plot the intercepts and join the points. Note the pattern of lines formed.
 Step 6: To obtain the solution we need to look for the intersection point. In this case there is no point of intersection as lines are parallel.(Recall that m_{1} = m_{2 }for parallel lines.)
 Step 7: Look at the graph below and relate the step from 1 to step 6.
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Let us look at three different examples to understand the topic graphically solving a pair of linear equations in detail with practical illustrations.
Solved Examples on Graphically Solving a Pair of Linear Equations

Example 1. Kia goes to a fair with $ 20 and wants to have the roller coaster rides and the dragon swing rides as well. The rides she took on the dragon swing are half the number of rides she took on roller coaster. If each roller coaster ride costs $3, and a dragon swing ride costs $ 4, how would you find out the number of rides she had on each, provided she spent the entire amount $20. Graphically solve the pair of linear equations.
Solution:
Let us denote the number of roller coaster rides that Kia had be x, and the number of dragon swing rides be y. Now the situation can be represented by the two equations:
y = (1/2) x
x  2y = 0 ......... (Eq. 1)
3x + 4y = 20............ (Eq. 2)To represent these equations graphically we need at least two solutions for each equation.
For 1st equation let us put x = 0, we get y = 0. Putting value 0 reduces the linear equation to one variable it is easy to solve.
Putting x = 2 gives y = 1.For equation 2, putting x = 0, we get
4y = 20 = y = 5.
Similarly putting y = 0 we get x = 20/3
Putting x = 4, we get y = 2.Below are the solutions given in the tabular form for both the equations.
Solutions for 1st equation
x 0 2 y = x/2 0 1 Solutions for 2nd equation,
x 0 20/3 4 y = (20  3x)/4 5 0 2 Look at the graph below representing the two equations graphically. Observe the pattern of two lines formed for two equations. They are intersecting at one point. Here in graph X axis denotes variable roller coaster rides and Y axis denotes dragon swing rides.
The two lines representing the two equations are intersecting at the point (4, 2). This means if the two lines are intersecting at the same point, then the point gives the unique solution for both the equations. 
Example 2. Ron purchased 2 pencils and 3 erasers for $9. His friend Sam purchased the same kind of 4 pencils and 6 erasers for $18. Form the equations. Graphically solve the pair of linear equations.
Solution:
Let us denote the cost of 1 pencil by $x and one eraser by $y.
Algebraic representation of the situation is given by the following equations:
2x + 3y = 9 ......(1)
4x + 6y = 18 ......... (2)To represent these equations graphically we need at least two solutions for each equation.
For 1st equation let us put x = 0, we get y = 3.
Putting x = 4.5 gives y = 0.For equation 2, putting x = 0, we get y = 3.
Similarly putting y = 1 we get x = 3Below are the solutions given in the tabular form for both the equations.
Solutions for 1st equation
x 0 4.5 y = (9  2x)/3 3 0 Solutions for 2nd equation,
x 0 3 y = (18  4x)/6 3 1 Look at the graph below representing the two equations graphically. Observe the pattern of two lines formed for two equations.
The two lines representing the two equations are coincident lines. This means if the two lines coincide, then there are infinitely many solutions. Each and every point on the line becomes a solution.
Here in graph X axis denotes pencil and Y axis denotes eraser. 
Example 3. Two metro routes are represented by equations, x + 2y – 4 = 0 and 2x + 4y – 12 = 0. Graphically solve the pair of linear equations.
Solution: To represent these equations graphically we need at least two solutions for each equation.
x + 2y – 4 = 0 ......(1)
2x + 4y – 12 = 0.......(2)For 1st equation let us put x = 0, we get y = 2.
Putting x = 4 gives y = 0.For equation 2, putting x = 0, we get y = 3.
Similarly putting x = 6 we get y = 0Below are the solutions given in the tabular form for both the equations.
Solutions for 1st equation
x 0 4 y = (4  x)/2 2 0 Solutions for 2nd equation,
x 0 6 y = (12  2x)/4 3 0 Look at the graph below representing the two equations graphically. Observe the pattern of two lines formed for two equations. To represent the equations graphically, we have 4 respective points S(0, 2) and T(4, 0) plot these points to
Here the two lines are parallel. The pair of equations is said to be inconsistent. In this case there is no solution.
get the line ST and the points R(0, 3) and P(6, 0) to get the line RP.
Practice Questions on Graphically Solving a Pair of Linear Equations
FAQs on Graphically Solving a Pair of Linear Equations
How do you Solve a Linear Equation Graphically?
To solve the linear equation graphically we need to find at least two solutions for the respective equations. After plotting the points graphically observe the pattern of lines to infer it is consistent, dependent or inconsistent. If the two lines are intersecting at the same point, then the point gives unique solution for both the equations. If the two lines coincide, then in this case there are infinitely many solutions. If the two lines are parallel, then in this case there is no solution.
What will be Solution of Two Pair of Linear Equation When the Equations are Solved Graphically?
When we solve the two pairs of linear equation graphically, after plotting the points geometrically we need to observe the pattern of lines. There could be three respective solutions according to the equations based on their representations. It could be consistent, dependent or inconsistent. If the two lines are intersecting at same same point, then the point gives unique solution for both the equations. If the two lines coincide, then in this case there are infinitely many solutions. If the two lines are parallel, then in this case there is no solution.
How do you Solve a Linear Equation with Two Variables?
To solve the linear equation with two variables we can use many methods. The listed below are the approved mathematical methods to solve the equations:
 Graphically
 Algebraically
 Cross Multiplication
 Substitution
 Elimination Method
How do you Describe Linear Equations?
Linear equations are the algebraic equations of first order which are of the form y = mx + b (involving two variables).
How do you Plot a Linear Equation in Two Variables?
 Find the solutions to the equation. Form a solution table for both the variables.
 Plot the points graphically and check the solutions. Observe the pattern of lines drawn through those points and obtain the solutions
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