Review the graphs of a system of two linear equations in two variables: x−5y=7 and 3x−2y=−4 . Find the solution to both equations

Answer 1

To find the solution to the system of equations given by:

1. $x-5y=7$
2. $3x-2y=-4$

we can use either the substitution method or the elimination method. Here, I'll use the substitution method.

Step 1: Solve the first equation for $x$

From the first equation:

$$x=5y+7$$

Step 2: Substitute for $x$ in the second equation

Now, substitute $x$ from the first equation into the second equation:

$$3(5y+7)-2y=-4$$

Step 3: Distribute and simplify

Distributing $3$:

$$15y+21-2y=-4$$

Combine like terms:

$$(15y-2y)+21=-4$$ $$13y+21=-4$$

Step 4: Isolate $y$

Subtract $21$ from both sides:

$$13y=-4-21$$ $$13y=-25$$

Now, divide both sides by $13$:

$$y=-\frac{25}{13}$$

Step 5: Substitute back to find $x$

Substitute $y$ back into the equation $x=5y+7$:

$$x=5(-\frac{25}{13})+7$$

Calculating $5\cdot -\frac{25}{13}$:

$$x=-\frac{125}{13}+7$$ $$x=-\frac{125}{13}+\frac{91}{13}$$ $$x=-\frac{125-91}{13}$$ $$x=-\frac{34}{13}$$

Final Solution

The solution to the system of equations is:

$$(-\frac{34}{13},-\frac{25}{13})$$

This means that the point $(-\frac{34}{13},-\frac{25}{13})$ is the point where the two lines intersect, which is the solution to the system.