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1.5 Linear Graphs – Slope Intercept Method

1.5.1 Prep

A. Solve the following equations for y so that each equation is in the form [latex]y=mx+b[/latex].

  1. [latex]2y=3x+8[/latex]
  2. [latex]-3y=3x-9[/latex]
  3. [latex]2x+3y=15[/latex]
  4. [latex]3x-3y=9[/latex]
  5. [latex]-y=2x+1[/latex]
  6. [latex]15y+5x=-15[/latex]
  7. [latex]6x-3y=18[/latex]
  8. [latex]\frac12x=\frac23y=4[/latex]
  9. [latex]\frac12x+2y=6[/latex]
  10. [latex]\frac{x+y}2=-3[/latex]
  11. [latex]\frac13x-2y=\frac23[/latex]
  12. [latex]\frac12x+=\frac34y=3[/latex]

B. Evaluate [latex]\frac{a-c}{b-d}[/latex] for the given values of a, b, c, and d.

  1. [latex]a=10,\;b=7;\;c=4,\;d=5[/latex]
  2. [latex]a=5,\;b=-7;\;c=-3,\;d=9[/latex]
  3. [latex]a=\frac12,\;b=\frac13;\;c=\frac52,\;d-\frac56[/latex]
  4. [latex]a=-\frac57,\;b=2;\;c=\frac23,\;d=\frac23[/latex]
  5. [latex]a=8,\;b=3;\;c=8,\;d=-2[/latex]
  6. [latex]a=-7,\;b=4;\;c=-1,\;d=4[/latex]
  7. When is having 0 in a fraction defined? When is it undefined? Discuss.

1.5.2 Preview

Slope-Intercept Form

When a linear equation in two variables is written in slope-intercept form

[latex]y = mx + b[/latex]

m is the slope of the line and (0, b) is the y-intercept of the line.

Example

To find the slope and y-intercept of a line, solve for y as shown in the following example.

[latex]3x+5y=10[/latex]

[latex]5y=-3x+10[/latex]

[latex]\frac{5y}5=\frac{-3x}5+\frac{10}5[/latex]

[latex]y=-\frac35x+2[/latex]

The slope is [latex]-\frac35[/latex] and the y-intercept is (0, 2).

Try It!

For the example above, write the reason for each step out to the side.

Try It!

Find the slope and y-intercept for [latex]-4x-3y=21[/latex].

The slope of a line is [latex]m=\frac{{\text{rise}}}{{\text{run}}}=\frac{{\text{change}}\;in\;y}{{\text{change}}\;in\;x}.[/latex]

A graph on a Cartesian coordinate system shows a straight black line passing through the origin with a positive slope. Two black dots are marked on the line, one in the third quadrant and one in the first quadrant. A dotted black right triangle is drawn connecting these two points, illustrating the 'rise' (vertical change) and 'run' (horizontal change) used to calculate the slope of the line. The 'rise' is labeled on the vertical dotted segment, and the 'run' is labeled on the horizontal dotted segment.

Try It!

Create possible values the rise and run show in the graph above label them on the graph.

Slope Formula

The slope m of the line containing the points (x1, y1) and (x2, y2) is given by

[latex]m=\frac{{\text{rise}}}{{\text{run}}}=\frac{{\text{change in y}}}{{\text{change in x}}}=\frac{y_2-y_1}{x_2-x_1}[/latex]

as long as [latex]x_2\neq x_1[/latex].

Example

The slope of the following line is [latex]m=\frac{{\text{down 3}}}{{\text{right 6}}}=\frac{-3}6=\frac{-1}2=\frac{{\text{down 1}}}{{\text{right 2}}}.[/latex]

A straight black line is plotted on a Cartesian coordinate system with a grid. The x-axis and y-axis range from -10 to 10 with integer markings. Two black dots are marked on the line. A dotted black right triangle is drawn connecting these two points, illustrating the slope. The vertical leg of the triangle has a length labeled '-3'. The horizontal leg of the triangle has a length labeled '6'.A straight black line with a negative slope is plotted on a Cartesian coordinate system with a grid. Four black dots are marked along the line. Dotted black right triangles are drawn connecting consecutive points, illustrating the slope. For each triangle, the vertical leg has a length labeled '-1', indicating a decrease of 1 unit in the y-direction, and the horizontal leg has a length labeled '2', indicating an increase of 2 units in the x-direction.

Try It!

  1. Draw two lines that are parallel.
  2. Draw two lines that are perpendicular.

Two nonvertical lines are parallel if they have the same slope but different y-intercepts.

Two nonvertical lines are perpendicular if the slope of one is the negative reciprocal of the slope of the other.

1.5.3 Classwork

A. Write the Slope-Intercept Form and identify the slope and y-intercept for each line below.

  1. [latex]2y=3x+8[/latex]
  2. [latex]-3y=3x-9[/latex]
  3. [latex]2x+3y=15[/latex]
  4. [latex]6x-3y=9[/latex]
  5. [latex]-3y=12[/latex]
  6. [latex]\frac12x=8[/latex]
  7. [latex]\frac13x-2y=\frac23[/latex]
  8. [latex]\frac12x-\frac23y=4[/latex]
  9. Write an equation for a line with slope [latex]\frac34[/latex]
  10. Write an equation for a line with y-intercept [latex](0,-50)[/latex]

B. Use the Slope-Intercept Method to graph the following lines.

  1. [latex]2x+3y=15[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  2. [latex]-5x+y=1[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  3. [latex]\frac58x-\frac14y=\frac12[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  4. [latex]\frac14x+\frac12y=-\frac23[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  5. Write this line in Slope-Intercept Form:[latex]-4x+y=0[/latex]. Find both the x- and y-intercepts. What can you do to find another point on the graph of the line?
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
    1. Plot the points [latex](6,-2)\;\text{and}\;(8,-3)[/latex], then draw the line that passes through them.
    2. Find the slope of this line by counting the rise over run between these two points.
      A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.

C. Find the slope of the line through the two points.

  1. [latex](6,-2)[/latex] and [latex](8,−3)[/latex]
  2. [latex]\left(\frac23,\frac56\right)[/latex] and [latex]\left(\frac53,-\frac76\right)[/latex]
  3. [latex](5,−3)[/latex] and [latex]\left(-2,-\frac38\right)[/latex]
  4. [latex]\left(1,\frac45\right)[/latex] and [latex]\left(\frac12,-\frac47\right)[/latex]
  5. [latex](5,−3)[/latex] and [latex](1,−3)[/latex]
  6. [latex](1, 4)[/latex] and [latex](1,−7)[/latex]

Parallel and Perpendicular Lines

  1. Use the following equations to answer the following.
    Line 1: [latex]y=-\frac25x+5[/latex]
    Line 2: [latex]2x+5y=15[/latex]
    1. Find the slope of Line 1.
    2. Find the slope of Line 2.
    3. What do you notice about their slopes, and what does it mean about their graphs?
  2. Use the following equations to answer the following.
    Line A: [latex]x-3y=1[/latex]
    Line B: [latex]y-3x+4=1-6x[/latex]
    1. Find the slope of Line A.
    2. Find the slope of Line B.
    3. What do you notice about their slopes, and what does it mean about their graphs?

1.5.4 Homework

A. Write the Slope-Intercept Form and identify the slope and y-intercept for each line below.

  1. [latex]\frac25x+\frac56y=10[/latex]
  2. [latex]x-3y=9[/latex]
  3. [latex]2x+2y=8[/latex]
  4. [latex]\frac14x-y=\frac23[/latex]
  5. [latex]\frac12x-\frac23y=5[/latex]
  6. [latex]\frac23y=-2[/latex]

B. Answer the following.

  1. Sketch the graphs of three different lines whose slopes are positive, negative, and zero.
  2. Is it possible for a line to have two different slopes? Why or why not?

C. Use the Slope-Intercept Method to graph the following lines.

  1. [latex]y=-\frac32x-5[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  2. [latex]3y=-x+9[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  3. [latex]-4y+x=12[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  4. [latex]\frac14y-\frac12x=\frac34[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  5. [latex]-6y=12[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.
  6. [latex]\frac12x-\frac23y=\frac52[/latex]
    A Cartesian coordinate plane is shown with a grid. The horizontal axis, labeled 'x', ranges from -10 to 10, with integer markings. The vertical axis, labeled 'y', also ranges from -10 to 10, with integer markings.

D. Find the slope of the line through the indicated points.

  1. Find the slope of the line through the points [latex](-1,4)[/latex] and [latex](5, 2)[/latex]
    1. using the slope formula.
    2. using the graph.
  2. [latex](4,3)[/latex] and [latex]\left(\frac72,\frac52\right)[/latex]
  3. [latex](-2,3)[/latex] and [latex](7,-8)[/latex]
  4. [latex](5,-3)[/latex] and [latex]\left(-2,\frac35\right)[/latex]
  5. [latex]\left(\frac56,-\frac35\right)[/latex] and [latex]\left(-2,\frac34\right)[/latex]

E. Solve the following.

  1. Which of the following pairs of lines are parallel or perpendicular?
    1. [latex]y=-\frac16x+11[/latex]
    2. [latex]y=6x-\frac12[/latex]
    3. [latex]x+6y=30[/latex]
    4. [latex]2x-12y=15[/latex]

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