3. Solve the following equations using the substitution method 3x - 4y = 10 and 5x + 8y = 2

To solve the system of equations using the substitution method, follow these steps:

### Given Equations:
1. \( 3x - 4y = 10 \)
2. \( 5x + 8y = 2 \)

### Step 1: Solve one equation for one variable
Let's solve the first equation for \( x \):

\[
3x = 4y + 10
\]

\[
x = \frac{4y + 10}{3}
\]

### Step 2: Substitute the expression for \( x \) into the second equation
Substitute \( x = \frac{4y + 10}{3} \) into the second equation \( 5x + 8y = 2 \):

\[
5\left(\frac{4y + 10}{3}\right) + 8y = 2
\]

### Step 3: Simplify and solve for \( y \)
Multiply both sides by 3 to eliminate the fraction:

\[
5(4y + 10) + 24y = 6
\]

Expand and combine like terms:

\[
20y + 50 + 24y = 6
\]

\[
44y + 50 = 6
\]

Subtract 50 from both sides:

\[
44y = -44
\]

Divide by 44:

\[
y = -1
\]

### Step 4: Substitute \( y \) back into the expression for \( x \)
Now, substitute \( y = -1 \) back into the expression for \( x \):

\[
x = \frac{4(-1) + 10}{3}
\]

\[
x = \frac{-4 + 10}{3}
\]

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

### Final Answer:
The solution to the system of equations is:

\[
x = 2, \quad y = -1
\]

Would you like any further details or have any questions about the solution?

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Here are 5 related questions for practice:

1. Solve the system \( x + 2y = 5 \) and \( 3x - y = 4 \) using the substitution method.
2. If \( 2x - 3y = 7 \) and \( x + 4y = 10 \), find \( x \) and \( y \) using substitution.
3. Solve \( 4x + 2y = 16 \) and \( 3x - y = 5 \) using substitution.
4. What is the solution to \( 7x - 5y = 9 \) and \( 2x + 3y = 1 \) using substitution?
5. Solve the system \( 3x + y = 0 \) and \( 4x - 2y = 8 \) using substitution.

**Tip:** When using the substitution method, always aim to express the variable in its simplest form before substituting it into the other equation. This makes the calculation easier.

Learn how to solve a system of equations using the substitution method with step-by-step guidance. This problem involves equations 3x - 4y = 10 and 5x + 8y = 2. Suitable for high school students.

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