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.

1 answer

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.