Question

At what point do the graphs of the two linear equations listed below intersect?

10x - 3y = 19
5x + 4y = -7
(-1, -0.5)
(1, -3)
(4, 7)
(-3, 2)

Answer 1

To find the point of intersection of the two linear equations, we need to solve the system of equations given by:

1. \( 10x - 3y = 19 \)
2. \( 5x + 4y = -7 \)

We can use either substitution or elimination to solve the system. Let's use the elimination method.

First, we can multiply the second equation by 2 to align the coefficients of \(x\) for elimination:

\[
2(5x + 4y) = 2(-7)
\]

This simplifies to:

\[
10x + 8y = -14
\]

Now we have a new system of equations:

1. \( 10x - 3y = 19 \)
2. \( 10x + 8y = -14 \)

Next, we can eliminate \(10x\) by subtracting the first equation from the second:

\[
(10x + 8y) - (10x - 3y) = -14 - 19
\]

This simplifies to:

\[
8y + 3y = -33
\]

Combining like terms, we get:

\[
11y = -33
\]

Dividing both sides by 11, we find:

\[
y = -3
\]

Now that we have \(y\), we can substitute this value back into one of the original equations to find \(x\). Using the second equation:

\[
5x + 4(-3) = -7
\]

This simplifies to:

\[
5x - 12 = -7
\]

Adding 12 to both sides:

\[
5x = 5
\]

Dividing both sides by 5 gives:

\[
x = 1
\]

Thus, the point of intersection of the two linear equations is:

\[
(x, y) = (1, -3)
\]

Therefore, the answer is \((1, -3)\).