Let's solve the system of equations where $y_1 = 5(2x - 8) - 2$ and $y_2 = 5(x - 3) + 3$, given that $y_1 = y_2$.

Step 1: Write down the equations

Given: $y_1 = 5(2x - 8) - 2$ $y_2 = 5(x - 3) + 3$

And we are told that $y_1 = y_2$.

Step 2: Substitute and set the equations equal

Since $y_1 = y_2$, we have: $5(2x - 8) - 2 = 5(x - 3) + 3$

Step 3: Simplify both sides

Expand the expressions on both sides: $5(2x - 8) - 2 = 10x - 40 - 2 = 10x - 42$ $5(x - 3) + 3 = 5x - 15 + 3 = 5x - 12$

So the equation becomes: $10x - 42 = 5x - 12$

Step 4: Solve for $x$

To solve for $x$, first get all terms involving $x$ on one side: $10x - 5x = -12 + 42$ $5x = 30$ $x = 6$

Step 5: Verify the solution

Substitute $x = 6$ back into the original expressions for $y_1$ and $y_2$:

1. For $y_1$: $y_1 = 5(2(6) - 8) - 2 = 5(12 - 8) - 2 = 5(4) - 2 = 20 - 2 = 18$

2. For $y_2$: $y_2 = 5(6 - 3) + 3 = 5(3) + 3 = 15 + 3 = 18$

Since both $y_1$ and $y_2$ equal 18 when $x = 6$, the solution is correct.

Final Answer:

The solution is $x = 6$.

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Would you like further details on any of the steps, or do you have other questions?

Here are 5 related questions for further practice:

1. Solve the system of equations: $y_1 = 3x + 7$ and $y_2 = 2x + 9$ when $y_1 = y_2$.
2. What is the solution for $y_1 = 4(3x - 2) + 5$ and $y_2 = 7(x + 1) - 3$ if $y_1 = y_2$?
3. How do you solve $y_1 = 6(2x + 3) - 4$ and $y_2 = 3x + 7$ with $y_1 = y_2$?
4. Solve the equation $8(2x - 5) + 3 = 4(x + 1) - 7$.
5. What is the value of $x$ when $7(3x - 4) + 9 = 5(2x + 3) - 8$?

Tip: When solving equations with multiple steps, always simplify the expressions fully before moving terms from one side of the equation to the other.