Question

solve each system of linear equations by graphing. 1. $y = \frac{3}{2}x - 1$ $y = -x + 4$ 2. $y = \frac{1}{3}x + 2$ $y = -\frac{4}{3}x - 3$ when you finish, the remaining letters will tell you the answer.

Explanation:

Step1: Solve System 1

We have the system of linear equations:

$$
LATEXBLOCK0
$$

Set the two equations equal to each other since they both equal \(y\):

$$ \frac{3}{2}x - 1=-x + 4 $$

Add \(x\) to both sides:

$$ \frac{3}{2}x+x - 1=4 $$

Combine like terms: \(\frac{3}{2}x+\frac{2}{2}x=\frac{5}{2}x\), so:

$$ \frac{5}{2}x - 1=4 $$

Add 1 to both sides:

$$ \frac{5}{2}x=5 $$

Multiply both sides by \(\frac{2}{5}\):

$$ x = 5\times\frac{2}{5}=2 $$

Substitute \(x = 2\) into \(y=-x + 4\):

$$ y=-2 + 4=2 $$

So the solution for system 1 is \((2,2)\).

Step2: Solve System 2

We have the system of linear equations:

$$
LATEXBLOCK1
$$

Set the two equations equal to each other:

$$ \frac{1}{3}x + 2=-\frac{4}{3}x-3 $$

Add \(\frac{4}{3}x\) to both sides:

$$ \frac{1}{3}x+\frac{4}{3}x + 2=-3 $$

Combine like terms: \(\frac{1 + 4}{3}x=\frac{5}{3}x\), so:

$$ \frac{5}{3}x+2=-3 $$

Subtract 2 from both sides:

$$ \frac{5}{3}x=-5 $$

Multiply both sides by \(\frac{3}{5}\):

$$ x=-5\times\frac{3}{5}=-3 $$

Substitute \(x = - 3\) into \(y=\frac{1}{3}x + 2\):

$$ y=\frac{1}{3}\times(-3)+2=-1 + 2=1 $$

So the solution for system 2 is \((-3,1)\).

Answer:

• System 1 solution: \((2,2)\)
• System 2 solution: \((-3,1)\)