Graph and then find the solution for y=1/2x+2 and y=-1/4x+8

1 answer

To find the solution for the two equations \(y = \frac{1}{2}x + 2\) and \(y = -\frac{1}{4}x + 8\), we can follow these steps:

  1. Graph the equations:

    • For the first equation, \(y = \frac{1}{2}x + 2\):

      • The y-intercept is 2 (which is the point (0, 2)).
      • The slope is \(\frac{1}{2}\), which means for every 2 units you move to the right (increase x by 2), you move up 1 unit (increase y by 1).
      • Another point can be found at \(x = 2\): \[y = \frac{1}{2}(2) + 2 = 1 + 2 = 3\] So another point is (2, 3).
    • For the second equation, \(y = -\frac{1}{4}x + 8\):

      • The y-intercept is 8 (which is the point (0, 8)).
      • The slope is \(-\frac{1}{4}\), meaning for every 4 units you move to the right (increase x by 4), you move down 1 unit (decrease y by 1).
      • Another point can be found at \(x = 4\): \[y = -\frac{1}{4}(4) + 8 = -1 + 8 = 7\] So another point is (4, 7).
  2. Plot the points and draw the lines:

    • Plot the points (0, 2) and (2, 3) for the first line and draw a line through them.
    • Plot the points (0, 8) and (4, 7) for the second line and draw a line through them.
  3. Find the intersection point:

    • To find the solution algebrically, set the two equations equal to each other: \[ \frac{1}{2}x + 2 = -\frac{1}{4}x + 8 \]

    • Multiply every term by 4 to eliminate the fractions: \[ 4 \left(\frac{1}{2}x\right) + 4(2) = 4\left(-\frac{1}{4}x\right) + 4(8) \] \[ 2x + 8 = -x + 32 \]

    • Add \(x\) to both sides: \[ 2x + x + 8 = 32 \] \[ 3x + 8 = 32 \]

    • Subtract 8 from both sides: \[ 3x = 24 \]

    • Divide by 3: \[ x = 8 \]

  4. Substitute \(x\) back into one of the original equations to find \(y\):

    • Using the first equation: \[ y = \frac{1}{2}(8) + 2 = 4 + 2 = 6 \]

So, the solution to the system of equations is \((8, 6)\).

Summary:

  • The intersection point is \((8, 6)\).
  • The lines representing the two equations intersect at this point.