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Wize High School Grade 10 Math Textbook > Systems of Linear Equations

Solving by Graphing

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Solving Graphically

One method for solving a system of linear equations is to first graph them, and visually see where the lines meet (cross over).



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Example
Solve the simultaneous equations 3x−y+2=0     1◯−2x−y−3=0     2◯\begin{aligned} 3x-y+2=0~~~~~\text{\textcircled 1}\\ -2x-y-3=0~~~~~\text{\textcircled 2} \end{aligned} .

Graphing

Let's first rearrange both equation into the slope y-intercept form y=mx+by=mx+b:

First equation: 3x−y+2=0−y+2=−3x−y=−3x−2−y−1=−3x−1−2−1y=3x+2\begin{array}{rcl} 3x-y+2&=&0\\[0.5em] -y+2&=&-3x\\[0.5em] -y&=&-3x-2\\[0.5em] \dfrac{-y}{-1}&=&\dfrac{-3x}{-1}-\dfrac{2}{-1}\\[1em] \bct y&\bct=&\bct{3x+2} \end{array}
  • y-intercept: y=2y=2
  • Slope: m=3 or 31m=3~\text{or}~\frac{3}{1} (up 3, right 1)

Second equation: −2x−y−3=0−y−3=2x−y=2x+3−y−1=2x−1+3−1y=−2x−3\begin{array}{rcl} -2x-y-3&=&0\\[0.5em] -y-3&=&2x\\[0.5em] -y&=&2x+3\\[0.5em] \dfrac{-y}{-1}&=&\dfrac{2x}{-1}+\dfrac{3}{-1}\\[1em] \bcfi y&\bcfi =&\bcfi{-2x-3} \end{array}
  • y-intercept: y=−3y=-3
  • Slope: m=−2 or −21m=-2~\text{or}~\frac{-2}{1} (down 2, right 1)

We get this graph:


Finding the Point of Intersection

From the graph, we see that the lines meet at the point (−1,−1)\boxed{(-1,-1)}.

Check Your Answer

We can confirm this by inserting the solution into the original equations.

Equation 1:
3x−y+2=03(−1)−(−1)+2=0−3+1+2=00=0  Confirmed!\begin{array}{rcl} 3x-y+2&=&0\\ 3\left(-1\right)-\left(-1\right)+2&=&0\\ -3+1+2&=&0\\ 0&=&0~\text{ Confirmed!} \end{array}
Equation 2:
−2x−y−3=0−2(−1)−(−1)−3=02+1−3=00=0  Confirmed!\begin{array}{rcl} -2x-y-3&=&0\\ -2\left(-1\right)-\left(-1\right)-3&=&0\\ 2+1-3&=&0\\ 0&=&0~\text{ Confirmed!} \end{array}

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Example: Solving Graphically

Find the solution to the system of linear equations which includes the following lines.

3x=61◯−6y−3x+6=02◯\begin{array}{cc} 3x=6&\text{\textcircled 1}\\ -6y-3x+6=0&\text{\textcircled 2} \end{array}

First put both in a form where the lines can be easily graphed.
  • 3x=6 3x=6\ becomes x=2\bct {x=2} (Note this is a vertical line so it is easily graphed)
  • −6y−3x+6 =0-6y-3x+6\ =0 becomes y=−12x +1\bcfi{y=-\frac{1}{2}x\ +1}

Next, graph the two lines.

Find the solution where the lines meet.
The solution is (2,0)\boxed{(2,0)}.

Practice: Solving Graphically

a) Graph the lines y=3x−5y=3x-5 and y=4y=4.

b) Determine the point of intersection between the lines y=3x−5y=3x-5 and y=4y=4.

Practice: Solving Graphically

Find the point of intersection between the lines y=2x +1y=2x\ +1 and y=−x+10y=-x+10

Practice: Solving Graphically

Find the point of intersection between the line y=23x +2y=\dfrac{2}{3}x\ +2 and the line perpendicular to it which also has a y-intercept of 2.