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1.23 Graphical solutions to linear equations or inequalities

Lesson

Solving linear equations graphically

Typically, we think of solving linear equation as a purely algebraic process. However, there is also a graphical interpretation of solving a linear equation. 

Consider the linear equation $2x+4=-6$2x+4=6, we often look at this and immediately start solving it as below.

$2x+4$2x+4 $=$= $-6$6
$2x+4-4$2x+44 $=$= $-6-4$64
$2x$2x $=$= $-10$10
$\frac{2x}{2}$2x2 $=$= $\frac{-10}{2}$102
$x$x $=$= $-5$5

We want to make connections in mathematics, so let's consider the connection between solving that linear equation and graphing two lines.

Suppose I wanted to find out where the lines $y=2x+4$y=2x+4 and $y=-6$y=6 intersect on the $xy$xy-plane. 

We could first graph $y=2x+4$y=2x+4 using the slope-intercept method, shown in blue. Then graph $y=-6$y=6 quickly as it is just a horizontal line through $y=-6$y=6, shown in green. We can see that these two lines intersect at the point $\left(-5,-6\right)$(5,6). In other words, the lines intersect where $x=-5$x=5 and $y=-6$y=6, which is consistent with the $x=-5$x=5 we got when solving algebraically.

 

 
 
Remember!

In general, if we have a linear equation like $ax+b=cx+d$ax+b=cx+d, then we can check the solution by graphing $y=ax+b$y=ax+b and $y=cx+d$y=cx+d and finding the x-coordinate of the point of intersection.

There will usually be one solution, but sometimes we can see none or an infinite number. 

 

Practice questions

Question 1

Consider the equation $\frac{3x}{5}-1=2$3x51=2.

  1. Solve for the value of $x$x that satisfies the equation.

  2. To verify the solution graphically, which two straight lines would need to be graphed?

    $y=\frac{3x}{5}$y=3x5

    A

    $y=\frac{3x}{5}+1$y=3x5+1

    B

    $y=2$y=2

    C

    $y=\frac{3x}{5}-1$y=3x51

    D
  3. Graph the lines $y=\frac{3x}{5}-1$y=3x51 and $y=2$y=2 on the same plane.

    Loading Graph...
  4. Hence find the value of $x$x that satisfies the two equations $y=\frac{3x}{5}-1$y=3x51 and $y=2$y=2 simultaneously.

Question 2

Consider the equation $2\left(x-1\right)-3=7$2(x1)3=7.

  1. Solve for the value of $x$x that satisfies the equation.

  2. To verify the solution graphically, which two straight lines would need to be graphed?

    $y=2\left(x-1\right)+3$y=2(x1)+3

    A

    $y=7$y=7

    B

    $y=2\left(x-1\right)-3$y=2(x1)3

    C

    $y=2\left(x-1\right)+4$y=2(x1)+4

    D
  3. Graph the lines $y=2\left(x-1\right)-3$y=2(x1)3 and $y=7$y=7 on the same plane.

    Loading Graph...
  4. Hence find the value of $x$x that satisfies the two equations $y=2\left(x-1\right)-3$y=2(x1)3 and $y=7$y=7 simultaneously.

Question 3

Using the graphs of $y=2x-7$y=2x7 and $y=-2x+1$y=2x+1, find the solution(s) of the equation $2x-7=-2x+1$2x7=2x+1.

 

Solving linear inequalities graphically

Similar to linear equations, we can also make a connection between linear inequalities and graphs of lines.

The key difference is that instead of just looking for where they intersect, but over what range of $x$x-values one line is above or below the other.

Previously, we learned how to solve linear inequalities. Remember that a linear inequality in $x$x will not involve $x^2$x2, $x^3$x3, $\frac{1}{x}$1x, or anything else. It will only ever involve $x$x being multiplied or divided by constants, or having constants added or subtracted.

In other words, the left hand side and right hand side of a linear equality will both be lines!

Think about it. If we have the linear inequality $3x-5<-5x+11$3x5<5x+11, the left hand side is the line $y_1=3x-5$y1=3x5 and the right hand side is the line $y_2=-5x+11$y2=5x+11.

We could solve the inequality algebraically like this.

$3x-5$3x5 $<$< $-5x+11$5x+11  
$8x-5$8x5 $<$< $11$11 Add $5x$5x to both sides
$8x$8x $<$< $16$16 Add $5$5 to both sides
$x$x $<$< $2$2 Divide both sides by $8$8.

Or, we could realize that to solve $3x-5<-5x+11$3x5<5x+11 we just need to find for what values of $x$x the line $y_1=3x-5$y1=3x5 is less than $y_2=-5x+11$y2=5x+11.

We can graph both lines like this.

For what values of $x$x is the line $y_1=3x-5$y1=3x5 less than $y_2=-5x+11$y2=5x+11? In other words, when is the line $y_1=3x-5$y1=3x5 below the line $y_2=-5x+11$y2=5x+11?

Just as we found by solving the inequality algebraically, $y_1$y1 is below $y_2$y2 for $x<2$x<2.

Worked examples

Question 4

Using the above graph, state the solution to the linear inequality $3x+9\ge0$3x+90 and plot the solution on a number line.

The solution to the inequality $3x+9\ge0$3x+90 will be whenever the line $y=3x+9$y=3x+9 is above the line $y=0$y=0 (the $x$x-axis), in other words, when the line is positive.

We can see that this happens when $x\ge-3$x3, which we can plot on the number line like this.

Question 5

Using the above graph, state the solution to the linear inequality $3x-21<9$3x21<9 and plot the solution on a number line.

 

The solution to the inequality $3x-21<9$3x21<9 will be whenever the line $y=3x-21$y=3x21 is below the line $y=9$y=9.

We can see that this happens when $x<10$x<10, which we can plot on the number line like this.

 

Practice questions

Question 6

Consider the graph of $y=x-6$y=x6.

Loading Graph...

  1. Using the graph, state the solution of the inequality $x-6<0$x6<0.

Question 7

Consider the graphs of $y=x+6$y=x+6 and $y=x-7$y=x7.

Loading Graph...

  1. Which of the following statements is true?

    The inequality $x+6\ge x-7$x+6x7 has one solution.

    A

    The inequality $x+6\ge x-7$x+6x7 has an infinite number of solutions.

    B

    The inequality $x+6\ge x-7$x+6x7 has no solutions.

    C

Question 8

To solve the inequality $x\le\frac{x-3}{4}-1$xx341, Tracy graphed $y=x-3$y=x3. What other line would she need to graph to be able to solve the inequality graphically?

  1. $y=4x+1$y=4x+1

    A

    $y=\frac{x-3}{4}-1$y=x341

    B

    $y=x$y=x

    C

    $y=4x+4$y=4x+4

    D

Outcomes

A.REI.D.11

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) =g(x); find the solutions approximately.

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