Question

What is the y value of the solution to the system of equations shown below?

{4x+3y=10
3x-2y=5

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Answer 1

To find the y-value of the solution to the system of equations given by:

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

we can solve this using substitution or elimination. Here, I'll use the elimination method.

First, let's rewrite both equations:

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

To eliminate \(y\), we can multiply Equation 1 by 2 and Equation 2 by 3:

1. \( 2(4x + 3y) = 2(10) \)
This simplifies to:
\( 8x + 6y = 20 \) (Equation 3)

2. \( 3(3x - 2y) = 3(5) \)
This simplifies to:
\( 9x - 6y = 15 \) (Equation 4)

Next, let's add Equation 3 and Equation 4 to eliminate \(y\):

\[
(8x + 6y) + (9x - 6y) = 20 + 15
\]

This simplifies to:

\[
8x + 9x + 6y - 6y = 35
\]

\[
17x = 35
\]

Now, solve for \(x\):

\[
x = \frac{35}{17}
\]

Now that we have \(x\), we can substitute it back into either of the original equations to find \(y\). Let's use Equation 1:

\[
4\left(\frac{35}{17}\right) + 3y = 10
\]

This simplifies to:

\[
\frac{140}{17} + 3y = 10
\]

Subtract \(\frac{140}{17}\) from both sides:

\[
3y = 10 - \frac{140}{17}
\]

Convert 10 to a fraction with a denominator of 17:

\[
10 = \frac{170}{17}
\]

So now:

\[
3y = \frac{170}{17} - \frac{140}{17}
\]

This simplifies to:

\[
3y = \frac{30}{17}
\]

Now, divide by 3:

\[
y = \frac{30}{17} \cdot \frac{1}{3} = \frac{10}{17}
\]

Therefore, the y-value of the solution to the system of equations is:

\[
\boxed{\frac{10}{17}}
\]