Linear Equations Review

Linear Equations Review

Module 4

Single Equations

Topics Covered

• Unit Rate

• Identifying slope, y intercept, and x intercept

• Graphing equations

• Writing equations from a graph

• Switching between slope-intercept form and standard form

Unit Rate

• Unit rate is the SAME as the slope of the line

• Time always goes in the denominator

• Example: making 40 sandwiches in 20 minutes

Multiple Choice

You make 60 sandwiches in 20 minutes. What is the unit rate of sandwiches per minute?

• 2

• 3

• 4

Multiple Choice

You make 60 sandwiches in 20 minutes. What is the slope?

• 2

• 3

• 4

Identifying Slope

• Positive if the line goes up from left to right

• Negative if the line goes down from left to right

• Counting boxes on the coordinate plane to calculate rise/run

Multiple Choice

Is this slope positive or negative?

• Positive

• Negative

Multiple Choice

Is this slope positive or negative?

• Positive

• Negative

Multiple Choice

What is the slope of this line?

 $slope=\frac{\text{rise}}{\text{run}}$  

•  $\frac{1}{4}$  

•  $-\frac{1}{4}$  
  

•  $\frac{4}{1}$  

•  $-\frac{4}{1}$  

Solving equations for y

• Get y by itself (shown in the next slide)

• Solving for y will give you an equation in the form y = mx + b

• Slope of the line is m (the coefficient on x)

• The y-intercept is b (the constant term)

In this equation, what is the slope? What is the y-intercept?

Graphing equations

• Pick two points, graph them, and connect the lines

• What to pick:

• The y-intercept is easy because in y = mx + b form, b is the y-intercept

• Then you can pick any other x value that will get you a whole number for y: if m is a fration, choose x = the denominator of m

X and Y intercepts

• x-intercept: What x is when y is 0? Where does the line cross the x-axis?

• y-intercept: What y is when x is 0? Where does the line cross the y-axis?

Finding the y-intercept

Finding the x-intercept

Graphing the intercepts: (0, -3) and (-4, 0)

Multiple Choice

What is the x-intercept? (set y=0)

 $4x-2y=-8$  

• -2

• 2

• 4

• -8

Multiple Choice

What is the y-intercept? (set x=0)

 $4x-2y=-8$  

• -2

• 2

• 4

• -8

Writing an equation in slope-intercept form from a graph

• Step 1: identify y-intercept

• Step 2: identify slope

• Step 3: write in the form y = mx + b

 $y=\frac{8}{2}x-3$  

 $y=-\frac{1}{4}x+6$  

Multiple Choice

What is the equation of this line?

•  $y=\frac{2}{3}x+4$  

•  $y=\frac{3}{2}x+4$  

•  $y=2x-3$  

•  $y=\frac{3}{2}x$  

Turning slope-intercept form into standard form

• Remember, slope-intercept form is y = mx + b and standard form is Ax + By = C

• Slope-intercept form has y by itself

• Standard form has x and y on the same side, and their coefficients must be whole numbers

Turning slope-intercept form into standard form

Multiple Choice

Convert to standard form:

 $y=-\frac{1}{4}x+5$  

•  $y+\frac{1}{4}x=5$  

•  $4y+x=5$  

•  $4y+x=20$  

Systems of Equations

Topics Covered

• Graphing systems of equations

• Determining how many solutions to a system of equations

• Solving a system of equations

• Writing a system of equations from a word problem

Graphing systems of equations

• Same techniques for graphing as with single equations

• The solution is the point (x, y) where the two lines cross

Multiple Choice

What is the solution? (x, y)

• (-3, 1)

• (1, -3)

• (1, 3)

How many solutions?

• No solutions: same slope, different y-intercepts

• Infinite solutions: same slope, same y-intercepts

• One solution: different slope, different or same y-intercepts

• In y = mx + b, slope is m and y-intercept is b

Multiple Choice

y = 6x + 3

y = -2x + 5

• No solutions

• Infinite solutions

• One solution

Solving 2 equations for their solution

1. Draw boxes around each place you see y OR an expression equivalent to y.

2. Replace y in one equation with the expression equivalent to y from the other equation.

3. Solve the resulting equation for x.

4. Replace x in one of the original equations with the x you solved for. Now, solve for y.

5. Your solution is the point (x, y)

A situation that doesn't require two substitutions

Point-Slope Form

• If you know the slope of the line and ANY point on the line, you can find the equation of that line

• If the point is

   $\left(x_1,\ y_1\right)$  

• And the slope is m

• The equation is  $y-y_1=m\left(x-x_1\right)$  

 $y-y_1=m\left(x-x_1\right)$  

• Example: point is (5, 0) and slope is 4

Multiple Choice

 $y-y_1=m\left(x-x_1\right)$  
The point is (3, 0) and the slope is 7

•  $y=7\left(x+3\right)$  

•  $y=7\left(x-3\right)$  

•  $y-3=7x$  

•  $y+3=7x$  

Word Problems

• Person A travels 120 miles in 2.5 hours

• Person B leaves 1/2 hour later and then travels 110 miles in 2 hours

• The solution represents the number of hours after person A left that they have traveled the same distance

Word Problems

• Person A travels 100 miles in 3 hours

• Person B leaves 1 hour later and then travels 95 miles in 2 hours

• The solution represents the number of hours after person A left that they have traveled the same distance

Word Problems

• At a store, apples cost $1 and oranges cost $2

• You buy 16 total fruits

• You spend $32

Word Problems

• At a store, apples cost $3 and oranges cost $0.80

• You buy 20 total fruits

• You spend $20.40