To find the solution to the system of equations given by \( y = 2x - 1 \) and \( y - 3 = 4x \), we can rewrite the second equation in slope-intercept form.
- Starting with the second equation: \[ y - 3 = 4x \] Add 3 to both sides: \[ y = 4x + 3 \]
Now we have two equations:
- \( y = 2x - 1 \)
- \( y = 4x + 3 \)
Next, set the two equations equal to each other to find the value of \( x \): \[ 2x - 1 = 4x + 3 \]
Now, solve for \( x \):
- Subtract \( 2x \) from both sides: \[ -1 = 2x + 3 \]
- Subtract 3 from both sides: \[ -4 = 2x \]
- Divide by 2: \[ x = -2 \]
Now that we have \( x = -2 \), we can substitute it back into one of the original equations to find \( y \). We'll use the first equation: \[ y = 2(-2) - 1 = -4 - 1 = -5 \]
The solution to the system of equations is the ordered pair \( (-2, -5) \).
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.