Question

fill in the blanks to explain one way to solve this system of equations using elimination.
$6x + 2y = 14$
$5x - 2y = 8$
first, add the two equations to get $11x = \square$. then substitute $\square$ for $x$ in either equation to get $y = \square$.

Explanation:

Step1: Add the two equations

We have the system of equations:

$$\begin{cases} 6x + 2y = 14 \\ 5x - 2y = 8 \end{cases}$$

When we add the left - hand sides of the two equations: \((6x + 2y)+(5x - 2y)=6x+5x + 2y-2y = 11x\)
When we add the right - hand sides of the two equations: \(14 + 8=22\)
So, after adding the two equations, we get \(11x = 22\)

Step2: Solve for x

From \(11x=22\), we can solve for \(x\) by dividing both sides of the equation by 11.
Using the formula \(x=\frac{22}{11}\), we find that \(x = 2\)

Step3: Substitute x = 2 into one of the original equations to solve for y

Let's substitute \(x = 2\) into the first equation \(6x+2y = 14\)
Substitute \(x = 2\) into \(6x+2y = 14\), we get \(6\times2+2y=14\)
Simplify the left - hand side: \(12 + 2y=14\)
Subtract 12 from both sides: \(2y=14 - 12=2\)
Divide both sides by 2: \(y=\frac{2}{2}=1\)

Answer:

First blank: 22, Second blank: 2, Third blank: 1