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  • Graphing Equations in Slope Intercept Form

    Let’s remember a linear equation can be written in:

    Standard Form: Ax + By = C

    It can also be written as:

    Slope-Intercept Form: y = mx + b (m = slope and b = y-intercept)

    To Graph:

    1. Plot the y-intercept (b in the equation).

    2. From the y-intercept, use the slope to tell us how to move:

    Example: m= → means from y-int. move UP 3 units and RIGHT 4 units.

    m= → means from y-int. move DOWN 2 units and RIGHT 3 units

    It is much easier to graph a line when the equation is in Slope-Intercept form.

    Write an equation in slope-intercept form for a line with a slope of and a y-

    intercept of -2. Then graph.

    Plug in what you know:

    y = mx + b

    𝑦 = 𝑥 − 2

    y-intercept = -2

    Slope means up 3 right 4

  • Write an equation in slope-intercept form for the line shown in the graph.

    To write an equation in slope-intercept form we need the slope and y-intercept.

    (Hint: you will need to use the slope formula and two points on the line to

    calculate slope).

    In the above graph we see that the y-intercept = 1 or (0, 1).

    Pick another point on the line (in this case, (2,0)).

    Calculate slope using these two points: 𝑚 = 0−1

    2−0 =

    −1

    2

    Put it all together:

    𝑦 = − 1

    2 𝑥 + 1

  • EXAMPLE: The band boosters are selling sandwiches for $5 each. They bought

    $1160 worth of ingredients. Write an equation for the profit made on x

    sandwiches and graph.

    𝑦 =5𝑥−1160

    It’s important to know that x cannot be negative, since x represents number of sandwiches.

    What would the profit be if 1400 sandwiches are sold (x=1400).

    𝑦=5(1400)−1160

    𝑦 =7000−1160

    𝑦 =5840

    The profit from selling 1400 sandwiches would be $5,840.

    - 1160

  • Writing Equations in Slope-Intercept Form

    Slope Intercept Form: y=mx + b (m = slope, b = y-int)

    If we are given:

    We can:

    Slope and coordinate

    A line has a slope of 1

    2 and

    passes through the point

    (2, 4)

    We can use the slope and coordinate to find y-int. (b):

    𝑦 =𝑚𝑥+𝑏 Substitute m, x and y (from coordinate) and solve for b.

    4=(2)( 1

    2 )+𝑏

    4=1+𝑏

    3=𝑏

    Now write the equation:

    𝑦 = 1

    2 𝑥+3

    Two coordinate points

    A line passes through the

    points (1, 2) and (3, 5)

    1. Calculate slope using slope formula

    𝑚 = 5 − 2

    3 − 1 =

    3

    2

    2. Now that we have the slope, we can use the procedure from above to find b. Just use one of the given points:

    2=(1)( 3

    2 )+𝑏

    1

    2 = 𝑏

    3. Write the equation 𝑦 = 3 2

    𝑥 + 1

    2

  • You Try! Write equations in slope-intercept form using the given information:

    Slope = -

    2

    3 and passes through (3, 1)

    𝑦 = − 2

    3 𝑥 + 𝑏

    1 = − 2

    3 (3) + 𝑏

    1 = −2 + 𝑏

    3 = 𝑏

    𝑦 = − 2

    3 𝑥 + 3

    Slope = 5 and passes through (0, 0)

    𝑦 = 5𝑥 + 𝑏

    0 = 5(0) + 𝑏

    0 = 𝑏

    𝑦 = 5𝑥

    Line that passes through (-3, 1) and (4, 5)

    𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1

    𝑚 =

    5 − 1

    4 + 3 =

    4

    7

    1 = 4

    7 (−3) + 𝑏

    1 = −12

    7 + 𝑏

    2 5

    7 = 𝑏

    𝑦 =

    4

    7 𝑥 + 2

    5

    7

    Line that passes through (0, -4) and (5, 5)

    𝑚 = 𝑦2 − 𝑦1 𝑥2 − 𝑥1

    𝑚 =

    5 + 4

    5 − 0 =

    9

    5

    −4 = 9

    5 (0) + 𝑏

    −4 = 𝑏

    𝑦 = 9

    5 𝑥 − 4

  • I can write linear equations in slope-intercept form given…..

    The slope and y-intercept

    Slope = 𝟏

    𝟑 y-int = -5

    𝒚 = 𝟏

    𝟑 𝒙 − 𝟓

    Standard Form

    4x – 2y = 14

    𝟒𝒙 − 𝟐𝒚 = 𝟏𝟒

    -𝟒𝒙 − 𝟒𝒙 −𝟐𝒚 = −𝟒𝒙 + 𝟏𝟒

    𝒚 = 𝟐𝒙 − 𝟕

    A graph

    m = −𝟐

    𝟓 b = -1

    𝒚 =

    −𝟐

    𝟓 𝒙 − 𝟏

    A point and a slope

    (-1, 3) and slope = -3

    𝒚 = 𝒎𝒙 + 𝒃

    𝟑 = −𝟑(−𝟏) + 𝒃 𝟑 = 𝟑 + 𝒃

    𝒃 = 𝟎 𝒚 = −𝟑𝒙

    Two points

    (-4, -7) and (8, -13)

    𝒎 = −𝟕 + 𝟏𝟑

    −𝟒 − 𝟖 =

    𝟔

    −𝟏𝟐 =

    −𝟏

    𝟐

    𝒚 = 𝒎𝒙 + 𝒃

    −𝟕 = − 𝟏

    𝟐 (−𝟒) + 𝒃

    −𝟕 = 𝟐 + 𝒃 𝒃 = −𝟗

    𝒚 = − 𝟏

    𝟐 𝒙 − 𝟗

  • Writing Equations in Point-Slope Form

    Remember:

    We’ve learned how to write linear equations in:

    Standard Form: Ax + By = C

    And

    Slope-Intercept Form: y = mx + b

    Today let’s write equations in Point-Slope Form. We use this form

    when we are given the slope of the line and one coordinate on the

    line.

    Point-Slope Form: 𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)

    where 𝑥1 𝑎𝑛𝑑 𝑦1 are given coordinates

    EXAMPLE:

    Write an equation in point-slope form for the line that passes through (3, -

    2) with a slope of .

    𝒚 − (−𝟐) = 1

    4 (𝒙 − 𝟑)

    𝒚 + 𝟐 = 1

    4 (𝒙 − 𝟑)

    NOTE: If given two points, use slope formula to get slope. Then use

    the slope and either of the given coordinates to write the equation

    in point-slope form.

  • Converting linear equations from one form to another:

    To convert from one form to another, use the rules of algebra to make

    the equation reflect the new form.

    To convert to Standard Form (Ax + By = C):

    Goal is to get the x term and y term on one side of the equal sign and

    the constant on the other. Remember the rules for A, B and C!

    To convert to Slope-Intercept Form (y = mx + b):

    Goal is to get the y alone on one side of the equal sign.

    YOU TRY!

    Convert to Standard Form:

    𝑦 − 1 = − (𝑥 − 5)

    𝑦 − 1 = −2

    3 𝑥 +

    10

    3

    2

    3 𝑥 + 𝑦 =

    13

    3

    2𝑥 + 3𝑦 = 13

    Convert to Slope-Int Form:

    𝑦 + 3 = (𝑥 + 1)

    𝑦 + 3 = 3

    2 𝑥 +

    3

    2

    𝑦 = 3

    2 𝑥 −

    3

    2

    Convert to Standard Form:

    𝑦 = 1

    2 𝑥 − 3

    − 1

    2 𝑥 + 𝑦 = −3

    𝑥 − 2𝑦 = 6

    Convert to Slope-Int Form:

    3𝑥 + 4𝑦 = 12

    4𝑦 = −3𝑥 + 12

    𝑦 = − 3

    4 𝑥 + 3

  • Parallel and Perpendicular Lines

    Parallel lines: lines that never meet.

    Perpendicular Lines: lines that cross and form right angles

    NOTE: Since both of these consist of two lines, there are two equations.

    Lines Rule Examples

    Parallel

    • The two lines have the SAME slope.

    • The two lines have DIFFERENT y – intercepts.

    𝑦=2𝑥+4 𝑦=2𝑥−3 𝑦=2𝑥

    All have slope of 2

    Perpendicular

    • The two lines have slopes

    that are OPPOSITE RECIPROCALS.

    • Any y-intercepts.

    𝑦=−3𝑥+6

    𝑦 = 1

    3 𝑥 − 2

    -3 and opposite reciprocals

    Intersecting

    • There is NO relationship

    between the slopes.

    • Any y-intercepts.

    𝑦=4𝑥+2 𝑦=−4𝑥−5

    4 and -4 are not the same and they are not opposite reciprocals

  • It is easiest to work with equations in slope-intercept form

    Steps in determining parallel and perpendicular lines or writing

    equations of parallel and perpendicular lines:

    Step 1: Convert the equation of the given line to slope-int form.

    Step 2: Determine slope of the new line.

    Step 3: Plug in what you know in y = mx + b to find the value of

    b Step 4: Write the equation of the new line.

    YOU TRY! Write an equation for the line that passes through (-3, 5) and is parallel to y = 2x – 4.

    Slope of given line = 2 therefore

    The slope of the second line = 2 (parallel- same) Find y-int (b) by using slope and coordinate:

    5 = 2(−3) + 𝑏 5 = −6 + 𝑏

    11 = 𝑏 Now write equation:

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