Question

What is the solution of y = 2x - 1 and y - 3 = 4x?
The ordered pair (__,__) is the solution to the system of equations because it is the point at which both lines intersect, so both lines share this point.

Answer 1

To find the solution to the system of equations given by \( y = 2x - 1 \) and \( y - 3 = 4x \), we can start by rewriting the second equation in slope-intercept form.

1. **Rewrite the second equation:**

\[
y - 3 = 4x
\]
Adding 3 to both sides:
\[
y = 4x + 3
\]

2. **Set the two equations equal to each other:**

Now we have:
\[
y = 2x - 1
\]
\[
y = 4x + 3
\]

We can set these equations equal to each other:
\[
2x - 1 = 4x + 3
\]

3. **Solve for x:**

Rearranging the equation:
\[
2x - 4x = 3 + 1
\]
\[
-2x = 4
\]
Dividing both sides by -2:
\[
x = -2
\]

4. **Substitute x back to find y:**

Now we can substitute \( x = -2 \) back into either equation to find \( y \). We'll use \( y = 2x - 1 \):
\[
y = 2(-2) - 1
\]
\[
y = -4 - 1
\]
\[
y = -5
\]

5. **Result:**

The ordered pair that is the solution to the system of equations is:
\[
(-2, -5)
\]

Thus, the ordered pair \((-2, -5)\) is the solution to the system of equations because it is the point at which both lines intersect, so both lines share this point.