Solve the system using elimination. 5x+4y=10 and 4x-y=8

Let's solve the system of equations using the elimination method:

\[
5x + 4y = 10 \quad \text{(Equation 1)}
\]
\[
4x - y = 8 \quad \text{(Equation 2)}
\]

### Step 1: Make the coefficients of one variable the same
We'll eliminate \( y \) by making the coefficients of \( y \) the same in both equations. To do this, we'll multiply Equation 2 by 4 to match the coefficient of \( y \) in Equation 1:

- Multiply Equation 2 by 4:
  \[
  4(4x - y) = 4(8)
  \]
  \[
  16x - 4y = 32 \quad \text{(Equation 3)}
  \]

Now, the system looks like this:

\[
5x + 4y = 10 \quad \text{(Equation 1)}
\]
\[
16x - 4y = 32 \quad \text{(Equation 3)}
\]

### Step 2: Add the equations to eliminate \( y \)
Now, add Equation 1 and Equation 3:

\[
(5x + 4y) + (16x - 4y) = 10 + 32
\]

Simplifying, we get:

\[
5x + 16x + 4y - 4y = 42
\]

\[
21x = 42
\]

### Step 3: Solve for \( x \)
Divide both sides by 21:

\[
x = \frac{42}{21}
\]

\[
x = 2
\]

### Step 4: Substitute \( x = 2 \) back into one of the original equations to find \( y \)
We'll substitute \( x = 2 \) into Equation 2:

\[
4(2) - y = 8
\]

Simplify:

\[
8 - y = 8
\]

Subtract 8 from both sides:

\[
-y = 0
\]

Multiply by \(-1\):

\[
y = 0
\]

### Solution
The solution to the system of equations is \( x = 2 \) and \( y = 0 \).

Would you like further details or have any questions? Here are some related questions you might consider:

1. How can you verify the solution to a system of equations?
2. What is the difference between consistent and inconsistent systems?
3. How do you graphically interpret the solution of this system?
4. Can elimination be used for nonlinear systems of equations?
5. How does the elimination method compare to using determinants?

**Tip:** When multiplying equations, make sure to distribute the multiplication to every term in the equation to avoid mistakes.

Learn how to solve the system of equations 5x + 4y = 10 and 4x - y = 8 using the elimination method with a detailed step-by-step solution. This problem covers systems of linear equations and the elimination method in algebra. Suitable for students in Grades 9-12.

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