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Jackson – Algebra 1 – Unit 5: Writing Linear Equations -- Notes

Unit 5 Calendar – Writing Linear Equations

Date Topic Homework HW √Nov 30 (A)Dec 1 (B)

Writing Linear Equations in Slope-Intercept Form/Using Linear Equations in Slope-Intercept Form

HW 1 Unit 5

Dec 2 (A)Dec 3 (B)

Writing Linear Equations in Point-Slope Form

Pg 305: 1-13 odd, 24, 26, 28 and 43

Dec 4 (A)Dec 7 (B)

Writing Linear Equations in Standard Form

Pg 314: 1-3, 11-25 odd, 31-35 odd, 39 a & b, 41 a, 43 a.

Dec 8 (A)Dec 9 (B)

Parallel and Perpendicular Lines

Pg 322: 1-15 odd, 19-25 odd, 29, 32 a, b and c.

Dec 10 (A)Dec 11 (B)

Lines of Best Fit/Predicting with Linear Models

Pg 328: 1-7, 14, 25, 26Pg 338: 1, 3-12

Dec 14 (A)Dec 15 (B)

Interpreting Linear Equations Activity/Review

Dec 16 (A)Dec 17 (B) Unit 5 Test None

Dec 18 (A)Line of Best Fit/Predicting with Linear Models Enrichment

None

Schedule Subject to Change!

Vocabulary: quadrants, coordinate plane, ordered pair, standard form of a linear equation, linear function, discrete function, continuous function, x-intercept, y-intercept, slope, rate of change, slope-intercept form, parallel, direct variation, function notation, family of functions, parent linear function

Textbook reference: Chapter 5

SOL A.7: The student will investigate and analyze function (linear) families and their characteristics both algebraically and graphically, including:

SOL A.7a: determining whether a relation is a function;SOL A.7b: domain and range;SOL A.7d: x- and y- intercepts;SOL A.7e: finding the values of a function for elements in its domain; andSOL A.7f: solve real-world problems involving equations.

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Chapter 5-1: Writing Linear Equations in Slope-Intercept Form

Reminder, slope-intercept form is

When you have the slope of the line and the y-intercept, you can write the linear equation in slope-intercept form.

Example.Write the linear equation in slope-intercept form.

The slope is 6, and the y-intercept is -9.

Practice.Write the linear equation in slope-intercept form.

The slope is -5/8, the y-intercept is 4. The slope is 10, the y-intercept is -17.

In word problems, often the y-intercept is the beginning value and the slope is the rate-of-change.

Example.There was already 7 inches of snow on the ground when it began snowing again. The snow was falling at a rate of 1/3 inch per hour.

Write an equation relating the total amount of snow on the ground (g) to the number of hours it had been snowing (h).

How much snow is on the ground after it has been snowing for 3 hours?

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Practice.A recording studio charges musicians an initial fee of $50 to record an album. Studio time costs an additional $35 per hour.

Write an equation that shows the total cost of recording an album (r) as a function of the studio time in hours (h).

How much does it cost to record an album if it takes 10 hours of studio time?

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Chapter 5-2: Using the slope formula to find the equation of a line.

Reminder: The slope formula is used to find the slope of a line if you are only given 2 points on the line.

When given the coordinates of ONE point AND the slope of the line, you can substitute the values into the slope formula.

We are given m so that will be substituted into the formula We are given one coordinate point, that that will be substituted into the formula for

(x1 , y1). x2∧ y2 remain our variables, but since we have only one set of unknown variables now,

they are simply x and y in our formula.

First, we can “rearrange” the slope formula:

m=y− y1x−x1

Multiply both sides by (x – x1) m (x−x1)= y− y1

Can also be written as y− y1=m(x−x1)

Now…we can substitute values. For example, when given the point (2, 4) and a slope of 8, substitute 2 for x1, 4 for y1, and 8 for m.

Substitute y−4=8(x−2)

Distribute y−4=8 x−16

Add 4 to both sides to isolate the y y=8 x−12

The equation is now in slope-intercept form (y =mx + b) y=8x−12

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Chapter 5-3: Point-Slope Form

Point-Slope Form enables us to write linear equations when we have the slope of the line and a point on that line.

y− y1=m(x−x1)

x∧ y are the equation variables m is the slope (x1 , y2) is any point on the line

Example.Write the equation of the line with a slope of 2 and goes through the point (3, 5).

Step 1. Substitute the given values into Point-Slope Form -- y− y1=m (x−x1)

y− y1=m(x−x1)

y−5=2(x−3)

Step 2. Distribute the slope. y−5=2 x−6

Step 3. Add the y1 value to both sides to present the equation in Slope-Intercept Form -- y=mx+b

y−5=2 x−6

+5+5

y=2x+1

Write the equation of the line, in slope-intercept form, that passes through the given point and has the given slope.

(-7, -5), m = -4 (8, -3), m = ½

Practice.Write the equation of the line, in slope-intercept form, that passes through the given point and has the given slope.

(0 ,1 ) ,m=2 (−6 ,−2 ) ,m=13

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What is the equation, in slope-intercept form, for the line with a slope of 3 that passes through the point (1, 5)?

Provide the equation, in slope-intercept form, of the line that passes through (-2, -7) and has a slope of 4?

A line has a slope of 112 and passes through the point (-6, -4), what is the equation of the line

in slope-intercept form?

Writing Linear Equations from Graphs

When given a graph, you can create the linear equation by identifying the y-intercept and the slope and entering the values into slope-intercept form (y = mx + b).

Example:

1) Identify the y –intercept(0, -2)

2) Determine the slope by countingGridlines (rise over run).

3/2

3) Enter values into equation.

y=32x−2

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Practice

Present the equations of each of the graphs in slope-intercept form.

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Chapter 5-4: Writing Linear Equations in Standard FormWe have been using Point-Slope Form to find the equation of a line given two points. We then solve for y to write the equation in Slope-Intercept Form.

Point-Slope:

To

Slope-Intercept:

The third form of linear equations is called Standard Form:

Where A, B and C are integers (no fractions or decimals).

*If you have fractions you need to multiply everything by a multiple of the denominator to get rid of the fractions.

We are “unisolating” the variable!

We are bringing x and y back together on the same side of the equation!

A is the coefficient of x. A must be positive and must be an integer (no fractions or decimals).

B is the coefficient of y. B can be positive or negative and must be an integer (no fractions or decimals).

C is a constant. C can be positive or negative and must be an integer (no fractions or decimals).

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Examples.

Write each equation in standard form.

1.

2.

3.

4. Write the standard form of an equation of the line passing through (-4, 3) with a slope of -2.

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Jackson – Algebra 1 – Unit 5: -- Notes

Practice.

Write each equation in standard form.

1. y=−7x+5 2. y+2=−65

(x−5)

3. y=34x+4 4.

12y+ 23x=−7

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Algebra Jackson Quarter 2/Unit 5

Chapter 5-5: Write Equations of Parallel and Perpendicular Lines

Two distinct lines in a coordinate plane either intersect or are parallel. Parallel lines are lines in the same plane that never intersect.

Nonvertical lines are parallel if they have the same slope and different y-intercepts.

The lines t and v are parallel lines.

Line t: y = ¾x + 4; Line v: y = ¾x – 2

They have the same slope (¾) and different y-intercepts.

Vertical lines are parallel if they have different x-intercepts. (Remember that the slope is undefined – VUX!).

The lines r and s are parallel lines.

Line r: x = -4; Line s: x = 6

They have the same slope (undefined) and

different x-intercepts.

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t

sr


Writing the equation of a line that is parallel to a given line.1. Identify the slope of the given line. The parallel line has the same slope.2. Enter the slope and point into point-slope form (y – y1 = m(x – x1)).3. Solve the equation for y to put into slope-intercept form (y = mx + b).

Example: A line passes through (12, 5) and is parallel to the graph of y = (2/3)x – 1.1. Identify the slope. Slope = 2/3

2. Enter into point-slope form. y – 5 = 2/3(x – 12)

3. Solve for y. y – 5 = (2/3)x – 8 +5 +5y = (2/3)x – 3

The graph of y = (2/3)x – 3 passes through (12, 5) and is parallel to the graph of y = (2/3)x – 1.

Practice

A line passes through (-3, -1) and is parallel to the graph of y = 2x + 3. What equation represents the line in slope-intercept form?

A line passes through (2, 8) and is parallel to the graph of y = -3x + 12. What equation represents the line in slope-intercept form?

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Perpendicular Lines

Two lines in a coordinate plane that intersect to form right angles are perpendicular. You can use slope to determine whether two lines are perpendicular.

Two nonvertical lines are perpendicular if the product of their slopes is -1. A vertical line and a horizontinal line are perpendicular.

Example:

The graph of y = ½x – 1 (line v) has a slope of ½

The graph of y = -2x +2 (line t) has a slope of -2

Since ½(-2) = -1, the lines are perpendicular.

Two numbers whose product is -1 are opposite reciprocals. Therefore, the slops of perpendicular lines are opposite reciprocals. To find the opposite reciprocal of a number, find its reciprocal; then find the opposite of the reciprocal.

For example, to find the opposite reciprocal of -3/4: Find the reciprocal => -4/3 Find its opposite => 4/3

Check: −34∙ 43=−1 4/3 is the opposite reciprocal of -3/4

Practice.

What is the slope of a line that is perpendicular to the graph of y = 5x – 2?

What is the slope of a line that is perpendicular to the graph of y = -½x + 12?

What is the slope of a line that is perpendicular to the graph of x = -4?

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Are the graphs of 4y = -5x + 12 and y=45x−8 parallel, perpendicular, or neither?

Are the graphs of y=34x+7 and 4 x−3 y=9 parallel, perpendicular, or neither?

Are the graphs of 6 y=−x+6 and y=−16x+6 parallel, perpendicular, or neither?

A line passes through (2, 4) and is perpendicular to the graph of y=13x+10. What equation

represents the line in slope-intercept form?

A line passes through (1, 8) and is perpendicular to the graph of y=2x+1. What equation represents the line in slope-intercept form?

A line passes through (3, 4) and is parallel to the graph y=5 x−8. What equation represents the line in slope-intercept form?

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Chapter 5-6: Line of Best Fit

Scatter Plots – a graph that shows the relationship between two variables.

When data is displayed with a scatter plot, it is difficult to analyze – the data is “scattered”.

Therefore, it is often useful to attempt to represent that data with the equation of a straight line in order to predict values that may not be represented by the scattered data.

The straight line is known as the Line of Best Fit and is also often referred to as a trend line.

The relationship between the variables can be determined by creating and analyzing the line of best fit.

The line of best fit is a line that has the same number of data elements above and below it and follows the trend of the graph.

Positive relationship – as one variable gets larger, so does the other.

Line of best fit = positive slope.

Negative relationship – as one variable gets larger, the other gets smaller.

Line of best fit = negative slope.

No Relationship – it is not clear if or how the data move together.

No apparent line of best fit.

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Sample DataSet A Set B Set C Set D

x (L1) y (L2) x (L1) y (L2) x (L1) y (L2) x (L1) y (L2)0 3 5 4 45 67 56 05 2.9 8 6 60 55 65 08 2.8 12 7 41 49 78 5

12.5 2.75 15 13 52 53 80 715 2.6 24 20 60 54 69 018 2.4 45 30 49 51 85 820 2.2 68 35 44 57 95 1535 1.8 95 40 46 51 88 1050 1 125 50 53 43 97 17

Enter Data into Calculator:1) STAT, Edit (1)2) Enter data in L1 (x values) and L2 (y values). Make sure the data is accurate and aligned.3) STATPlot (2nd + Y=). With cursor on Plot 1, ENTER. Make sure “On” is highlighted, and

that xlist: L1 and ylist: L2

4) ZOOM, ZoomStat (9)5) This displays the Scatter Plot.

Calculating Line of Best Fit – using data that you have already entered.1) STAT. Toggle to CALC and select LineReg (ax+b) (4).2) Enter the list where you entered your data. The default is L1,L2 so if that is where your

data is stored, you do not need to change anything.3) Scroll to Calculated and ENTER.4) This is your line of best fit in Slope-intercept form: y = ax + b

a = slope b = y-intercept

What is the line of best fit for data set A? What is the line of best fit for data set B?

What is the line of best fit for data set C? What is the line of best fit for data set D?

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Chapter 5-7 Predict with Linear Models: The Zeroes of a Function

Reminder: The x-intercept of a function is the point where the graph of the function crosses the x-axis.

From an equation, the x-intercept is found by setting y = 0 and solving for x.

In function notation, this is saying that f(x) = 0.

The ZERO of a function is the x-coordinate of the x-intercept.

1) From graphs:a) b)

x-intercept:______ x-intercept:______

zero:______ zero:______

2) From equations: (Remember, substitute 0 for y and solve for x)

a) y = 5x-20 b) f(x) = ¾x + 6

x-intercept:______ x-intercept:______

zero:______ zero:______

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