 y1 =5(2x-8) -2, y2 =5(x-3) +3, and y1 =y2


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 \( x = 6 \).

---

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.

Learn how to solve the system of equations 5(2x-8)-2 = 5(x-3)+3 with a detailed step-by-step solution. This problem covers linear equations and the substitution method. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation