Wize High School Grade 9 Math Textbook > Analytic Geometry: Equation of a Line

Intro to Systems of Linear Equations
A system of linear equation or simultaneous equations is when you have more than 1 linear equation. In this course, we will see systems with only 2 linear equations.

What does "Solving a System of Linear Equations" Mean?
Solving a system of linear equations means we want to find both xxx and yyy numbers that "fit" into both equations.

Since linear equations look like straight lines in a graph, we are actually finding the point (x, y)\left(x,\ y\right)(x, y) where the two lines cross over. This is called the point of intersection.


There are a few special cases where the lines do not meet (no solution) or where the lines are the same (many solutions).

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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}3x−y+2=0     1◯−2x−y−3=0     2◯​ .

Graphing

Let's first rearrange both equation into the slope y-intercept form y=mx+by=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}3x−y+2−y+2−y−1−y​y​=====​0−3x−3x−2−1−3x​−−12​3x+2​
y-intercept: y=2y=2y=2
Slope: m=3 or 31m=3~\text{or}~\frac{3}{1}m=3 or 13​ (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}−2x−y−3−y−3−y−1−y​y​=====​02x2x+3−12x​+−13​−2x−3​
y-intercept: y=−3y=-3y=−3
Slope: m=−2 or −21m=-2~\text{or}~\frac{-2}{1}m=−2 or 1−2​ (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)}(−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}3x−y+23(−1)−(−1)+2−3+1+20​====​0000  Confirmed!​
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}−2x−y−3−2(−1)−(−1)−32+1−30​====​0000  Confirmed!​

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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}3x=6−6y−3x+6=0​1◯2◯​

First put both in a form where the lines can be easily graphed.
3x=6 3x=6\ 3x=6 becomes x=2\bct {x=2}x=2  (Note this is a vertical line so it is easily graphed)
−6y−3x+6 =0-6y-3x+6\ =0−6y−3x+6 =0 becomes y=−12x +1\bcfi{y=-\frac{1}{2}x\ +1}y=−21​x +1

Next, graph the two lines.

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

Practice: Solving Graphically
a) Graph the lines y=3x−5y=3x-5y=3x−5 and y=4y=4y=4.

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

b) Enter the point of intersection in the form

(x,y)

I don't know

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

Enter your answer in the form

(x,y)

I don't know

Practice: Solving Graphically
Find the point of intersection between the line y=23x +2y=\dfrac{2}{3}x\ +2y=32​x +2 and the line perpendicular to it which also has a y-intercept of 2.

Enter your answer in the form

(x,y)

I don't know