Solve: $$ 7 x-8 y=24 $$ (a) $$ -10 x+16 y=-16 $$

Question

UpStudy AI · Accepted Answer

To solve the system of equations:

$$
\begin{cases}
7x - 8y = 24 \quad \text{(Equation 1)} \
-10x + 16y = -16 \quad \text{(Equation (a))}
\end{cases}
$$

we can use the **elimination method**. Here's a step-by-step solution:

### Step 1: Simplify Equation (a)

First, simplify Equation (a) by dividing all terms by 2 to make the coefficients smaller:

$$
-10x + 16y = -16 \quad \Rightarrow \quad -5x + 8y = -8 \quad \text{(Equation 2)}
$$

Now, the system of equations is:

$$
\begin{cases}
7x - 8y = 24 \quad \text{(Equation 1)} \
-5x + 8y = -8 \quad \text{(Equation 2)}
\end{cases}
$$

### Step 2: Add Equations to Eliminate $$ y $$

Add Equation 1 and Equation 2 to eliminate $$ y $$:

$$
(7x - 8y) + (-5x + 8y) = 24 + (-8)
$$

Simplifying:

$$
7x - 5x - 8y + 8y = 16 \quad \Rightarrow \quad 2x = 16
$$

### Step 3: Solve for $$ x $$

Divide both sides by 2:

$$
x = \frac{16}{2} = 8
$$

### Step 4: Substitute $$ x = 8 $$ into One of the Original Equations

Let's substitute $$ x = 8 $$ into Equation 1:

$$
7(8) - 8y = 24 \quad \Rightarrow \quad 56 - 8y = 24
$$

Subtract 56 from both sides:

$$
-8y = 24 - 56 \quad \Rightarrow \quad -8y = -32
$$

Divide both sides by -8:

$$
y = \frac{-32}{-8} = 4
$$

### Final Solution

$$
x = 8 \quad \text{and} \quad y = 4
$$

**Answer:** The solution is $$ x = 8 $$ and $$ y = 4 $$.
The solution is $$ x = 8 $$ and $$ y = 4 $$.

Solve: \( 7 x-8 y=24 \) (a) \( -10 x+16 y=-16 \)