Question

solving systems of equations by graphing
use a graph to solve each system of equations. list the solution.

1. {y = 2x - 1

y = -4x - 7

1. {18x - 3y = 21

y = 6x - 7

1. {y = 6x + 4

6x - y = 1
use a graph to approximate the solution of each system. list the estimated solution.

Explanation:

Step1: Recall the concept

To solve a system of linear - equations by graphing, we graph each equation in the same coordinate plane. The solution is the point of intersection of the two lines.

Step2: For the first system

$$\begin{cases}y = 2x-1\\y=-4x - 7\end{cases}$$

The first equation $y = 2x-1$ has a y - intercept of $b=-1$ and a slope of $m = 2$. The second equation $y=-4x - 7$ has a y - intercept of $b=-7$ and a slope of $m=-4$.
We set $2x-1=-4x - 7$.
Add $4x$ to both sides: $2x + 4x-1=-4x+4x - 7$, which simplifies to $6x-1=-7$.
Add 1 to both sides: $6x-1 + 1=-7 + 1$, so $6x=-6$.
Divide both sides by 6: $x=-1$.
Substitute $x = - 1$ into $y = 2x-1$, then $y=2\times(-1)-1=-2 - 1=-3$.

Step3: For the second system

$$\begin{cases}18x-3y=21\\y = 6x-7\end{cases}$$

Rewrite the first equation $18x-3y=21$ in slope - intercept form $y=mx + b$.
First, subtract $18x$ from both sides: $-3y=-18x + 21$.
Divide by $-3$: $y = 6x-7$.
Since the two equations are the same, the system has infinitely many solutions.

Step4: For the third system

$$\begin{cases}y=6x + 4\\6x-y=1\end{cases}$$

Rewrite the second equation $6x-y=1$ in slope - intercept form: $y=6x - 1$.
The first equation $y = 6x+4$ has a y - intercept of $b = 4$ and a slope of $m = 6$. The second equation $y=6x - 1$ has a y - intercept of $b=-1$ and a slope of $m = 6$.
Since the slopes are the same and the y - intercepts are different, the lines are parallel and there is no solution.

Answer:

1. $(-1,-3)$
2. Infinitely many solutions
3. No solution