Linear Equations

Which of the following points are y-intercepts? Select all that apply. a. (0, 9) b. (-7, 0) c. (0, -3) d. (2, 0)

(0, 9) and (0, -3)

In the graph y=2x+14, where is the y-intercept?

(0,14). Found by making x=0.

In the graph y=2x+14, where is the x-intercept?

(-7,0). Found by making y=0.

What are the coordinates of the y-intercept for the line y = 2x - 3? a. (-3, 0) b. (3, 0) c. (0, -3) d. (0, 3)

(0, -3)

The run is found by subtracting x-values

(x1, y1) (x2, y2) run = x2 - x1

On a graph, the slope measures the steepness of a line

- A line with a positive slope goes up from left to right. - A line with a negative slope goes down from left to right.

Consider the ordered pairs (3, -2), (-1, 5), and (-9, -8). Which of the following is NOT an x-coordinate? a. -8 b. 3 c. -1 d. 6

-8

To graph a linear equation in two variables:

1.) Find three ordered pairs that are solutions to the linear equation. 2.) Plot the ordered pairs on the graph. 3.) Draw a line through the three points.

Which of the following equations shows an INCORRECT substitution of the ordered pair (3, 5)? a. -2x - y = 0 -2(3) - 5 = 0 b. 9x = y + 2 9(3) = 5 + 2 c. 3y = 2x + 1 3(3) = 2(5) + 1 d. -4 = 9y + 3x -4 = 9(5) + 3(3)

3y = 2x + 1 3(3) = 2(5) + 1

Which of the following equations is written in standard form? a. 5x = 3y + 4 b. y = 3x - 5 c. 4x + 3y = 5 d. 6y = 2 + 4x

4x + 3y = 5

(2, 3)

Find the solution.

(2, 4)

Find the solution.

(2, 6)

Find the solution.

(2, 8)

Find the solution.

(3, 2)

Find the solution.

(3, 6)

Find the solution.

(4, 2)

Find the solution.

Graphing vertical lines

If an equation may be written in the form x = a, where a is a real number, then the graph of the equation is a vertical line; that is, it is a straight line that is parallel to the y-axis

Graphing horizontal lines

If an equation may be written in the form y = b, where b is a real number, then the graph of the equation is a horizontal line; that is, it is a straight line that is parallel to the x-axis

Most linear equations have one x-intercept and one y-intercept.

Some linear equations only have one intercept.

Which of the following is best described by rise / run? a. The slope-intercept form of a line b. The x-intercept of a line c. The y-intercept of a line d. The slope of a line

The slope of a line

y = -2x + 3

What is the equation of the line shown?

What is the slope of a horizontal line?

Zero.

The slope-intercept form of an equation of a line that has a slope m and a y-intercept (0, b) is given by the formula y = mx + b

in slope-intercept form, the m is only the coefficient of x, the variable part is not included in the slope.

Ordered Pair

is a solution in an equation if it results in a true statement when the values for the x-coordinate and the y-coordinate are substituted into the equation.

Standard Form

is an equation that can be written in the form Ax + By = C, where A, B, and C are real numbers, but A and B are not both zero.

One way that we can describe the slope of a line on a graph is

m = rise / run

Which of the following best describes the slope m of a line? a. m = run / rise b. slope = change in x-coordinates / change in y-coordinates c. m = y / x d. m = rise / run

m = rise / run

Which formula is the best choice for finding the slope of a line when two points on the line are given? a. Ax + By = C b. y - y1 = m(x - x1) c. y = mx + b d. m = y2 - y1 / x2 - x1

m = y2 - y1 / x2 - x1

Which of the following formulas is used to find the slope for a line passing through the two points (x1, y1) and (x2, x1)? a. m = y2 - x2 / y1 - x1 b. m = x / y c. m = y2 - y1 / x2 - x1 d. m = x2 = x1 / y2 - y1

m = y2 - y1 / x2 - x1

In the graph y=2x+14, what is the slope?

m=2.

Which of the following does NOT appear on a blank graph or in the definition of the rectangular coordinate system? a. ordered pair b. y-axis c. origin d. x-axis

ordered pair

To write the equation of a line, you need to find these two things......

slope (m) and y-intercept (b)

y = 4 - 6x (What form is this?)

slope intercept form

The m in the y=mx+b is the

slope of the line

Intercept

the point at which a line crosses an axis is called an intercept.

On the x-axis

the positive direction is to the right and the negative direction is to the left.

Write the equation if b = -2 and m = 3

y = 3x - 2

Which of the following is the slope-intercept form of a line? a. y = bx + m b. m = y2 - y1 / x2 - x1 c. Ax + By = C d. y = mx + b

y = mx + b

The b in y=mx+b is the

y-intercept (where the line crosses the y-axis)

Write the equation if b = 3 and m = -2

y= -2x+3

Slope-intercept form for equation of a line.

y=mx+b

Is (3, 5) a solution to 3x + 2y = 19?

3(3) + 2(5) = 19 9 + 10 = 19 19 = 19 TRUE

Is (6, -4) a solution to 3x = 2y + 10?

3(6) = 2(-4) + 10 18 = -8 + 10 18 = 2 FALSE

Which of the following statements about the y-intercept is FALSE? a. To find the y-intercept, let x = 0 and solve for y b. In the equation y = mx + b, the y-intercept is represented by the letter b c. In the equation y = mx + b, the y-intercept is the y-value d. The y-intercept is the point where the line crosses the y-axis

In the equation y = mx + b, the y-intercept is the y-value

In the equation y = mx + b, which letter represents the slope? a. the slope is x b. the slope is y c. the slope is m d. the slope is b

the slope is m

x-Intercept

the x-intercept of a line is the point where the line crosses the x-axis. The x-intercept is an ordered pair with the coordinates (a, 0), where a is a real number.

b

the y-intercept

Which of the following is NOT a solution to the equation x - y = 5? a. (4, -1) b. (9, 4) c. (0, 5) d. (8, 3)

(0, 5)

Slope

the rate of change between two ordered pair solutions to a linear equation is called a slope. The variable m is typically used to represent slope.

Rise

the rise of a line is the change in the vertical position, or the difference between the y-coordinates of two points on the line.

Run

the run of a line is the change in the horizontal position, or the difference between the x-coordinates of two points on the line.

y-Intercept

the y-intercept of a line is the point where the line crosses the y-axis. The y-intercept is an ordered pair with the coordinates (o, b), where b is a real number.

(-2, -2)

Find the solution.

(0, 0)

Find the solution.

(0, 3)

Find the solution.

(2, 20)

Find the solution.

The rise is found by subtracting y-values

(x1, y1) (x2, y2) rise = y2 - y1

Graphing a linear equation with slope and y-intercept

1.) Find the slope and the y-intercept. 2.) Plot the y-intercept on the graph. 3.) Begin at the y-intercept. Move up or down the number of units represented by the numerator of the slope (rise). If the numerator is positive, move up. If the numerator is negative, move down. 4.) From that position, move left or right the number of units represented by the denominator of the slope (run). If the denominator is positive, move right. If the denominator is negative, move left. 5.) Plot this point and draw a line that passes through this point and the y-intercept. This will give you the graph of the line.

To graph an equation using the intercepts:

1.) Find the x-intercept by letting y = 0 and solving for x. 2.) Find the y-intercept by letting x = 0 and solving for y. 3.) Plot the two points on a graph. 4.) Draw a line through the pints. Put an arrow on each end of the line to indicate that the line extends in both directions.

Finding the y-intercept of a linear equation:

1.) Let x = 0 2.) Solve for y

Finding the x-intercept of a linear equation:

1.) Let y = 0 2.) Solve for x

To find a solution to a linear equation in two variables:

1.) Substitute a value for one of the variables. 2.) Solve the equation to find the value of the other variable.

Which of the following is the CORRECT method for using the slope and y-intercept to graph a line? a. Find three points that are solutions by substituting values for x and/or y b. Beginning at the origin, use the slope to find the first point, then use the y-intercept to graph an additional point c. Plot the y-intercept first, then use the slope to find an additional point d. The slope and y-intercept cannot be used to graph a line

Plot the y-intercept first, then use the slope to find an additional point

What must be TRUE about the slope in order for it to be used to graph a line? a. Slop must be in fraction form or rewritten as a fraction b. The slope cannot be negative c. You must know the equation of the line first d. The slope-intercept equation must be solved for m

Slope must be in fraction form or rewritten as a fraction

Which is the correct process for plotting the ordered pair (-2, -1)? a. Start at the origin, move two units to the right, and then move up one unit. b. Start at the origin, move two units down, and then move one unit to the left. c. Start at the origin, move two units to the left, and then move down one unit. d. Start at the origin, move one unit down, and then move two units to the right.

Start at the origin, move two units to the left, and then move down one unit.

Which of the following does NOT describe the x-intercept of a line? a. The x-intercept of a line is the point where the line crosses the x-axis. b. The x-intercept of a line is found by letting x = 0 and solving for y. c. If a linear equation has an x-intercept, it can be one of the points used to graph a line. d. The x-intercept of a line is an ordered pair with the coordinates (a, 0), where a is a real number.

The x-intercept of a line is found by letting x = 0 and solving for y.

What is the slope of a vertical line?

Undefined.

y = 3x - 2

What is the equation of the line shown?

Origin

is represented by the ordered pair (0, 0), and it has both x- and y-coordinates of 0.

x-Coordinate

is the first number in an ordered pair and it represents the location relative to the x-axis.

y-Coordinate

is the second number in an ordered pair and it represents the location relative to the y-axis.

On the y-axis

the positive direction is up and the negative direction is down.

Which of the following is the CORRECT slope-intercept form for a line with slope m = -1/2 and y-intercept (0, 4)? a. y = 1/2x - 4 b. y = 4x - 1/2 c. 1/2x + y = 4 d. y = -1/2 + 4

y = -1/2 + 4

Choose the equation that will be a horizontal line when graphed. a. y = 2 b. x - y = 0 c. x = 4 d. y = 5x

y = 2