We are given a system of two linear equations: $5x + 3y = 8$ $4x + 7y = 34$ We need to explain why the same pair of $x$ and $y$ values that satisfy these two equations also satisfy the equation $9x + 10y = 42$.

1. Problem Description

We are given a system of two linear equations:

$5x + 3y = 8$

$4x + 7y = 34$

We need to explain why the same pair of $x$ and $y$ values that satisfy these two equations also satisfy the equation $9x + 10y = 42$.

2. Solution Steps

First, let's solve for $x$ and $y$ in the given system of equations. We can use elimination or substitution. Let's use elimination.

Multiply the first equation by 4 and the second equation by 5:

$4(5x + 3y) = 4(8) \Rightarrow 20x + 12y = 32$

$5(4x + 7y) = 5(34) \Rightarrow 20x + 35y = 170$

Subtract the first modified equation from the second modified equation:

$(20x + 35y) - (20x + 12y) = 170 - 32$

$23y = 138$

$y = \frac{138}{23} = 6$

Now, substitute $y = 6$ into the first equation:

$5x + 3(6) = 8$

$5x + 18 = 8$

$5x = 8 - 18$

$5x = -10$

$x = -2$

So the solution to the system of equations is $x = -2$ and $y = 6$.

Now let's check if these values satisfy the equation $9x + 10y = 42$:

$9(-2) + 10(6) = -18 + 60 = 42$

Since $9x + 10y = 42$ holds true for $x = -2$ and $y = 6$, we need to explain why.

Notice that if we add the original two equations, we get:

$(5x + 3y) + (4x + 7y) = 8 + 34$

$9x + 10y = 42$

Therefore, the equation $9x + 10y = 42$ is simply the sum of the two equations $5x + 3y = 8$ and $4x + 7y = 34$.

Since the solution $(x, y)$ satisfies both equations, it must also satisfy their sum.

3. Final Answer

The equation $9x + 10y = 42$ is the sum of the equations $5x + 3y = 8$ and $4x + 7y = 34$. Therefore, any solution that satisfies the first two equations will also satisfy the third.