what is the solution for y=y=-4/3x-6 and y=4/3x+2

To find the solution for the system of equations:

1. \( y = -\frac{4}{3}x - 6 \)
2. \( y = \frac{4}{3}x + 2 \)

We solve them by setting the two equations equal to each other since both are equal to \( y \):

\[
-\frac{4}{3}x - 6 = \frac{4}{3}x + 2
\]

### Step 1: Combine like terms

To get rid of the fractions, first move all terms involving \( x \) to one side. Add \( \frac{4}{3}x \) to both sides:

\[
-\frac{4}{3}x + \frac{4}{3}x - 6 = \frac{4}{3}x + \frac{4}{3}x + 2
\]

This simplifies to:

\[
-6 = \frac{8}{3}x + 2
\]

### Step 2: Isolate the variable

Next, subtract 2 from both sides:

\[
-6 - 2 = \frac{8}{3}x
\]

\[
-8 = \frac{8}{3}x
\]

### Step 3: Solve for \( x \)

Multiply both sides by the reciprocal of \( \frac{8}{3} \), which is \( \frac{3}{8} \):

\[
x = -8 \times \frac{3}{8} = -3
\]

### Step 4: Solve for \( y \)

Now substitute \( x = -3 \) back into one of the original equations to solve for \( y \). Using the first equation:

\[
y = -\frac{4}{3}(-3) - 6
\]

\[
y = 4 - 6 = -2
\]

### Solution:

The solution to the system of equations is \( (x, y) = (-3, -2) \).

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Would you like more details on any of the steps? 

Here are some related questions:

1. What is a system of linear equations?
2. How do you determine if a system of equations has one solution, no solution, or infinitely many solutions?
3. What are other methods for solving systems of equations (e.g., substitution, elimination)?
4. How can you graphically represent the solution to a system of equations?
5. What happens if the slopes of the lines in a system are equal?

**Tip**: When solving systems of linear equations, always check the slopes first; parallel lines have no solutions.

Find the solution to the system of linear equations y = -4/3x - 6 and y = 4/3x + 2 through a step-by-step process. This algebra problem involves key concepts like linear equations and solving systems by setting equations equal to each other. Ideal for Grades 8-10.

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