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Put the equation into slope-intercept form:

2y + 4x = 12

5x - y = 9

What form is this equation in?

What do you know in this form?

Can you graph it easily?

Find the slope of a line perpendicular to each given line.
What does a perpendicular slope look like?

Y = -1/2x - 2

y + 7 = 5/2 (x + 6)

What if we wanted a parallel line? What does the slope of a parallel line look like?


Find the slope, y-intercept, and x-intercept for each equation.

y - 4 = 5(x + 2)

8x - 7y = 17

1 answer

1. Convert the equation into slope-intercept form

Equation:
\[ 2y + 4x = 12 \]

Steps to convert to slope-intercept form (y = mx + b):

  1. Subtract \(4x\) from both sides:
    \[ 2y = -4x + 12 \]

  2. Divide each term by 2:
    \[ y = -2x + 6 \]

Slope-intercept form:
\[ y = -2x + 6 \]

2. Equation: \( 5x - y = 9 \)

What form is this equation in?
This equation is in standard form (Ax + By = C).

What do you know in this form?
In the standard form, you can easily find intercepts, and it clearly shows the relationship between the coefficients (A, B, C).

Can you graph it easily?
It’s not as straightforward as slope-intercept form, as you first need to rearrange it to find the slope and y-intercept.

Convert the equation to slope-intercept form:

  1. Add \(y\) to both sides:
    \[ 5x - 9 = y \]

  2. Rewrite:
    \[ y = 5x - 9 \]

3. Find the slope of a line perpendicular to each given line

To find the perpendicular slope, take the negative reciprocal of the original slope.

For \(y = -2x + 6\):

  • The slope \(m = -2\).
  • The perpendicular slope \(m_{\perpendicular} = \frac{1}{2}\).

For \(y = 5x - 9\):

  • The slope \(m = 5\).
  • The perpendicular slope \(m_{\perpendicular} = -\frac{1}{5}\).

4. What does a perpendicular slope look like?

A perpendicular slope is obtained by flipping the fraction of the slope and changing its sign.

5. Given two equations:

  • Equation 1:

    \[ y = -\frac{1}{2}x - 2 \]

    • Slope \(m = -\frac{1}{2}\)
    • Perpendicular slope \(m_{\perpendicular} = 2\)

  • Equation 2:

    \[ y + 7 = \frac{5}{2}(x + 6) \]

    • Rewrite to slope-intercept form:
      \[ y + 7 = \frac{5}{2}x + 15 \]
      \[ y = \frac{5}{2}x + 8 \]
    • Slope \(m = \frac{5}{2}\)
    • Perpendicular slope \(m_{\perpendicular} = -\frac{2}{5}\)

6. What if we wanted a parallel line?

The slope of a parallel line is the same as the original line.

Examples of parallel slopes:

  • For the line \(y = -2x + 6\), a parallel line will also have a slope of \(-2\).
  • For the line \(y = 5x - 9\), a parallel line will also have a slope of \(5\).
  • For the line \(y = -\frac{1}{2}x - 2\), a parallel line will also have a slope of \(-\frac{1}{2}\).
  • For the line \(y = \frac{5}{2}x + 8\), a parallel line will also have a slope of \(\frac{5}{2}\).

7. Find the slope, y-intercept, and x-intercept for each equation

  1. Equation:
    \[ y - 4 = 5(x + 2) \]

    Convert to slope-intercept form: \[ y - 4 = 5x + 10 \]
    \[ y = 5x + 14 \]

    • Slope (m): 5
    • y-intercept (b): 14 (at (0, 14))
    • x-intercept: Set \(y = 0\):
      \[ 0 = 5x + 14 \]
      \[ 5x = -14 \]
      \[ x = -\frac{14}{5} \] (at \((-2.8, 0)\))

  1. Equation:
    \[ 8x - 7y = 17 \]

    Convert to slope-intercept form: \[ -7y = -8x + 17 \]
    \[ y = \frac{8}{7}x - \frac{17}{7} \]

    • Slope (m): \(\frac{8}{7}\)
    • y-intercept (b): \(-\frac{17}{7}\) (at \((0, -\frac{17}{7})\))
    • x-intercept: Set \(y = 0\):
      \[ 0 = 8x - 17 \]
      \[ 8x = 17 \]
      \[ x = \frac{17}{8} \] (at \((\frac{17}{8}, 0)\))

These steps provide you with all the information regarding the slopes, intercepts, and how to determine parallel and perpendicular lines.