Graphing Linear Equations Review - Week 1

Linear Equations Review

What is a Linear Equation?

An equation whose graph is a line.

Linear Graph

• An equation whose graph is a line is a linear equation.

• You can plot points on a coordinate plane to graph a linear equation.

• Example: y = -4x + 4.

X and Y Intercepts

• Linear equations have x and y-intercepts. 

• The x-intercept is the x-coordinate of the point at which the graph crosses the x-axis.

• The y-intercept is the y-coordinate of the point at which the graph crosses the y-axis. 

Multiple Choice

Which of the following shows a linear equation?

A linear equation can be shown 3 ways:

• xy table

• graph

• equation

Multiple Choice

What are the 3 ways to show a linear equation?

• Graph

• X | Y Table

• Equation

• All of the above

Evaluating Linear Equations

What does it mean to evaluate?

Evaluating Linear Equations

• Replace the variable with its value

• Follow the Order of Operations

• You will find the answer - the value of the whole equation

Quick
Review

Example

Multiple Choice

Solve 3(2)

• 6

• 5

• 1

• 4

Dropdown

3(2) means 3​

Select answer

2

Dropdown

12(5) means 12 ​

Select answer

5

Multiple Choice

Solve 3 ( 12 - 7)

• 15

• 29

• 8

• 5

Fill in the Blank

What is the value of this expression?

$$50-\left(3\times5\right)+\left(2\times4\right)$$

Multiple Choice

Which statement is true about the parentheses in this expression?

$$36\ \div(2\times2)+20$$

• The parentheses indicate that $$36\div2$$ should be solved first

• The parentheses indicate that

  $$2+20$$ should be solved first

• The parentheses indicate that $$2\times2$$ should be solved first

• The parentheses indicate that $$2\times2$$ should be solved last

Multiple Choice

(6 + 2) x (3 + 2)

• 18
• 26
• 16
• 40

Practice Evaluating Linear Equations

Fill in the Blank

y = 5x - 3 when x = 2

Fill in the Blank

y = -2x + 13 when x = 6

Fill in the Blank

y = 3x + 10 when x = -5

-

It is going to allow us to determine the inputs (x-values) and outputs (y-values) for our X | Y table. We can use this table to graph on a coordinate plan and see patterns.

Why do we need to know how to evaluate linear equations?

In your notebook, make an XY Table!

• Evaluate each x value

• That number will go in the y box next to its x value

Evaluate y = 2x + 1 when x = -1

Evaluate y = 2x + 1 when x = 0

Evaluate y = 2x + 1 when x = 1

Evaluate y = 2x + 1 when x = 2

Fill in the Blank

Evaluate y = 2x + 1 when x = 2

Here's the complete XY Table

Notice the pattern?!?

The values on the XY Table can become ordered pairs.

Ordered pairs are shown in the format (x, y)

Ta-Da! The graph is a straight line!

If you don't get a straight line, you made a mistake somewhere in your calculation or graphing.

Graphing

In your notebook, create this table and complete it by evaluating the linear equation. Graph the line on the coordinate plane.

Important Characteristics of Linear Equations

1. Linear Equation: An equation between two variables that creates a straight line when graphed.
2. Slope: The measure of the steepness of a line. It represents the ratio of the vertical change to the horizontal change between any two points on the line. (How tilted the line is)
3. X or Y Intercept: The point where a line crosses the x-axis or the y-axis.
4. Solution: A value or set of values that make an equation true. (If we plug in coordinate points (x,y) into a linear equation and it is true, then the point is a solution to the linear equation.)
5. Coefficient: The numerical value that multiplies a variable in a term. In a linear equation, it represents the slope of the line.
6. Variable: A symbol representing an unknown quantity in an equation. In linear equations, commonly used variables are x and y.
7. Constant: A number that remains the same in an equation. In a linear equation, the value is not multiplied by a variable.
8. Graph: A visual representation of data or an equation on a coordinate plane. In linear equations, it shows the relationship between the variables as a line.

In order to determine of an ordered pair is a solution, we must check to see if they are on the graph. If they are on the graph then they are a solution. If we could not see the graph, we can plug in the x and y values to see if the statement is true.

Are they solutions?

​Substitute each coordinate point above into the linear equation to confirm they are solutions.

​The math word decided it is RISE on top and RUN on bottom
EVERY SINGLE TIME
so we are all on the same page.

Slope from a graph

There are four types of slopes.

Positive

Negative

Zero

Undefined

Positive Slopes

We read slopes like we read a book, left to right.

Positive slopes go UP as you go from left to right on a graph.

Negative Slopes

We read slopes like we read a book, left to right.

Negative slopes go DOWN as you go from left to right on a graph.

Zero Slopes

We read slopes like we read a book, left to right.

Zero slopes go STRAIGHT ACROSS (HORIZONTAL) as you go from left to right on a graph.

Undefined Slopes

We read slopes like we read a book, left to right.

Undefined slopes go STRAIGHT UP AND DOWN (VERTICAL) as you go from left to right on a graph.

Multiple Choice

What type of slope?

• zero

• undefined

• negative

• positive

Multiple Choice

What type of slope?

• zero

• undefined

• negative

• positive

Multiple Choice

What type of slope?

• zero

• undefined

• negative

• positive

Multiple Choice

Find the slope of the line.

• Undefined
• 0
• 1
• -1

Multiple Choice

Find the slope:

• 3/4
• 4/3
• -3/4
• -4/3

Multiple Choice

What is the slope of :

• $$m=\frac{1}{3}$$

• $$m=3$$

• $$m=-\frac{1}{3}$$

• $$m=-3$$

Multiple Choice

Find the slope of the line.

• 1/2
• -1/2
• 2/1
• -2/1

Multiple Choice

What type of slope?

• zero

• undefined

• negative

• positive

Use the slope formula.
Remember: Y values on top of X values

To find slope using 2 points

Multiple Choice

find the slope of the line passing through

(2, 5) and (3, 6)

• Undefined

• m = 0

• m = -1

• m = 1

Multiple Choice

Find the slope of the line passing through

(-4, 5) and (-8, -5)

• $$m\ =\frac{5}{2}$$

• $$m=\frac{2}{5}$$

• $$m=0$$

• undefined

• $$m=-\frac{2}{5}$$

Multiple Choice

Find the slope of the line that passes through the points (2, 4) and (6, 12)

• 1/2
• -1/2
• 2
• -2

Multiple Choice

Find the slope of the line that passes through
(10, 1) and (5, 2)

• 1/5
• -1/5
• 5
• -5

Multiple Choice

What is the slope?

• -1/3
• -3
• -4
• 5

Points that touch the x-axis or y-axis.
x-intercept touches the x-axis
y-intercept touches the y-axis

Intercepts

Multiple Choice

What are the x- and y -intercepts on this graph?

• (0,5), (0,5)
• (5,0), (5,0)
• (5,0), (0,5)
• (0,0), (5,0)

Multiple Choice

What is the x-intercept for the given graph?

• 1
• -2
• 2
• 1/2

Multiple Choice

^{What is the x- intercept of the line?}

• x = 1
• x = 2
• x = -1
• x = -2

Multiple Choice

What is the x-intercept shown on the graph?

• (2, 0)
• (-2, 0)
• (0, 3)
• (3, 0)

How to find intercepts using the linear equation

You will evaluate the linear equation twice.

First, you will set y equal to 0 to calculate the x-intercept.


Second, you will set x equal to 0 to calculate the y-intercept.

Multiple Choice

Give the x-intercept
3x + 8y = 24

• (8, 0)
• (0, 8)
• (3, 0)
• (0, 3)

Multiple Choice

Give the y-interecept
3x + y = 5

• -3
• (0,5)
• -5
• (0, -3)

Multiple Choice

What is the y intercept?
y = 2x - 3

• 3
• -3
• 1
• 2

Multiple Choice

What is the y-intercept in the equation x + 2y = 4

• 0
• 2
• 4
• KC Royals

Multiple Choice

Determine the y-intercept for the following equation
4x + 2y = -12

• 4
• -6
• -12
• 2

Multiple Choice

Which ordered pair could represent a y-intercept on a graph?

• (6, 0)
• (0, 4)
• (-1, 3)
• (2, 0)

There are 3 ways to write
THE SAME
linear equation

• ​Standard Form is a general form for working with linear equations but may not be the most convenient for graphing directly.

• Slope-Intercept Form is excellent for graphing because it makes the slope and y-intercept easily identifiable, allowing for a quick sketch of the line.

• Point-Slope Form is useful when you have a specific point and the slope and need to find the equation for the line passing through that point, which can then be used for graphing.

Multiple Choice

$$y=mx+b\$$  is what form of a linear equation.

• Point-Slope Form

• Slope-Intercept Form

• Standard  Form

Multiple Choice

$$Ax+By=C$$   is what form of a linear equation. 

• Point-Slope Form

• Slope-Intercept Form

• Standard Form

Multiple Choice

$$y-y_1=m\left(x-x_1\right)$$   is what form of a linear equation. 

• Point-Slope Form

• Slope-Intercept Form

• Standard Form

Multiple Choice

The equation  $$3x+5y=7$$   is in what form?

• Point-Slope Form

• Slope-Intercept Form

• Standard Form

Multiple Choice

The equation  $$y=\frac{1}{2}x+3$$   is in what form?

• Point-Slope Form

• Slope-Intercept Form

• Standard Form

Multiple Choice

The equation  $$y-3=\frac{1}{3}\left(x-5\right)$$   is in what form?

• Point-Slope Form

• Slope-Intercept Form

• Standard Form

Multiple Choice

What does "m" represent in y=mx+b?

• y-intercept
• slope
• macaroni
• nothing

Multiple Choice

What is the slope in the equation:  y=4x+3

• 3
• 4x
• 4
• x

Multiple Choice

What is the y-intercept in the following equation:  y=-4x-5

• -4
• -4x
• 5
• -5

Multiple Choice

How would you write the equation with a slope of 2/3 and a y-intercept of -3

• y=3/2x+3
• y=2/3-3
• y=2/3x-3
• y=2/3x+3

Multiple Choice

If m=3 and b=6, what is the correct slope intercept equation?

• y=3-6
• y=3x-6
• y=3x+6
• y=6x+3

Multiple Choice

Write the equation for the line in slope-intercept form.

• y= -1x+3
• y= 4x+3
• y=1/4x+3
• y= -4x+3

Drag and Drop

The y-intercept is ​

Placeholder

and the slope is ​ ​

Placeholder

Drag these tiles and drop them in the correct blank above

(0,-3)

(-3,0)

(4,0)

(0,4)

Drag and Drop

The y-intercept is ​

Placeholder

and the slope is ​

Placeholder

Drag these tiles and drop them in the correct blank above

(0,-3)

(-3,0)

(4,0)

(0,4)

​Slope Intercept Form

​y = mx + b

​slope

​y - intercept

Drag and Drop

Given:  $$y=-\frac{1}{3}x+5$$ , the slope is ​

Placeholder

and the y-intercept is ​

Placeholder

Drag these tiles and drop them in the correct blank above

5

-5

Drag and Drop

Given:  $$y=\frac{3}{5}x$$ , the slope is ​

Placeholder

and the y-intercept is ​

Placeholder

Drag these tiles and drop them in the correct blank above

0

1

Multiple Choice

Which graph represents the function:   $$y=-\frac{1}{2}x-2$$  

Multiple Choice

Which graph represents the function:   $$y=\frac{3}{2}x+3$$  

EXAMPLE

What is the equation of the line graphed?

y = mx + b



Multiple Choice

What equation is graphed?

• $$y=5x+1$$

• $$y=-5x+1$$

• $$y=\frac{1}{5}x+1$$

• $$y=-\frac{1}{5}x+1$$

Math Response

Write the equation of the line in Slope-Intercept Form. ( $$y=mx+b$$ )

Type answer here

Correct Answer: y=-\frac{3}{5}x-3

Deg^°

Rad

Math Response

Write the equation of the line in Slope-Intercept Form. ( $$y=mx+b$$ )

Type answer here

Correct Answer: y=\frac{1}{2}x-2

Deg^°

Rad