Sec 2-2 Linear Equations in Two Variables

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)

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)

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.

Which point could NOT be used to graph the linear equation y = 2x - 4? a. (2, 0) b. (3, 3) c. (1, -2) d. (0, -4)

(3, 3)

The rise is found by subtracting y-values

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

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.

When describing perpendicular lines, m1×m2=?

-1

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

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.) Find one additional ordered pair solution so that you have three points with which to plot the line. 4.) Plot the three points on a graph. 5.) Draw a line through the pints. Put an arrow on each end of the line to indicate that the line extends in both directions.

Writing equations in slope-intercept form: Slope = 2 y-intercept = (0, 4/3) y = mx + b y = 2x + 4/3

1.) Find the y-intercept. 2.) Find the slope m = y2 - y1 / x2 - x1 3.) Substitute the values into the slope-intercept form y = mx + b y = mx + b y = 2x + 4/3

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.

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

Finding the slope of a line that passes through (x1, y1) and (x2, y2)

1.) Put it into the formula m = y2 - y1 / x2 - x1 2.) Write down point-slope form y - y1 = m(x - x1) 3.) Substitute in m, x, and y 4.) Simplify 5.) Solve for y to get slope-intercept form y = mx + b

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.

The point-slope form of an equation of a line whose slope is m and that passes through the point (x1, y1) is given by y - y1 = m (x - x1)

1.) Write the point-slope equation y - y1 = m (x - x1) 2.) Substitute the given values of x, y, and m into the equation 3.) To write the equation in slope-intercept form, solve for y; y = mx + b

Which of the following equations will have MORE than one intercept? a. x = 3 b. 2x + y = 1 c. -2x + 5y = 0 d. y = 2x

2x + y = 1

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 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

Graphing linear equations through the origin

An equation in the form Ax + By = 0, where A and B are not both zero, will pass through the origin and the only intercept will be (0, 0)

Which would NOT be a step in solving 2x + 3y = -7, when x = 1? a. Write the solution as an ordered pair (1, -3) b. Substitute x = 1 into the equation and simplify to 2 + 3y = -7 c. Divide each side by -3 to result in y = 3 d. Subtract 2 from each side to result in 3y = -9

Divide each side by -3 to result in y = 3

What would be the LEAST useful step to graphing 2x - y = 0? a. Create a table of values. b. Find both the x- and y-intercepts. c. Use the ordered pair (0, 0) d. Find three solutions to the equation.

Find both the x- and y-intercepts.

Quadrants are labeled as follows:

I = RUQ = Right Upper Quadrant II = LUQ = Left Upper Quadrant III = LLQ = Left Lower Quadrant IV = RLQ = Right Lower Quadrant

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

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

Which of the following is FALSE about the equation x + 5 = 0? a. It is the graph of a vertical line. b. It can be re-written as x = -5 c. It has one y-intercept. d. The x-value is always -5

It has one y-intercept.

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

NOTE:

Points that lie on an axis are not considered to be in any quadrant.

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

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

Some linear equations only have one intercept.

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 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

The slope of a horizontal line is 0.

The slope of a vertical line is undefined. It can also be said to have "no slope."

Given the points (2, 2) and (4, 6), which of the following is TRUE? a. The slope of the line between the points is -2. b. The slope of the line between the points is 0. c. The slope of the line between the points is 2. d. The slope of the line between the points is undefined.

The slope of the line between the points is 2.

Which of the following is TRUE for the point-slope form of a linear equation, y - y1 = m(x - x1)? a. The slope of the line is zero b. The slope of the line is m and the y-intercept is (0, y1) c. The slope of the line is m and the line passes through the ordered pair (x1, y1) d. There is no y-intercept

The slope of the line is m and the line passes through the ordered pair (x1, y1)

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.

Which of the following is TRUE? a. The y-intercept can be found without any calculations when the equation is written in slope-intercept form b. In order to write the equation of a line given two points, one of the points must have an x- or y-coordinate of 0 c. It is best to use the standard form of the equation of a line when given two points d. When a line is in the form of Ax + By = C, the values of A, B, or C can never equal 0

The y-intercept can be found without any calculations when the equation is written in slope-intercept form

What is the slope of a vertical line?

Undefined.

What if the point given is not the y-intercept?

We can use another form of a linear equation called point-slope form.

Where do you find the value for x?

You simply choose it ;-)

What is the slope of a horizontal line? a. Positive b. Zero c. Undefined d. Negative

Zero

What is the slope of a horizontal line?

Zero.

When solving linear equations in two variables that contains a fraction,

choose multiples of the denominator to end up working with integers.

For the ordered pair (3, -4), in which direction would the coordinate -4 be plotted, and by how many units? a. right four units b. left four units c. down four units e. up four units

down four units

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.

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.

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

Remember the formula to find slope is

m = y2 - y1 / x2 - x1

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

Lines that have the same slopes are?

parallel

Lines that have negative reciprocals

perpendicular

y + 9 = 4(x - 3) (What form is this?)

point slope form

If I know a random point and a slope, which form would the easiest to use to write the equation?

point slope form is best to use when you know the slope and point that is not the y-intercept

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

slope intercept form

If I know the slope and y-intercept, which form of a linear equation should I use?

slope intercept works best when you know slope and the y-intercept

5x + 2y = 12 (What form is this?)

standard form

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.

On the y-axis

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

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.

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

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.

Making a table of values can help keep the ordered pairs organized.

x + y ---------- -4 | 0 -5 | 1 3 | -7

Find the x- and y-intercepts of 3x - 6y = 12

x = (4, 0) and y = (0, -2) 3x - 6(0) = 12 3x - 0 = 12 Divide both sides by 3 x = 4 3(0) - 6y = 12 -6y = 12 Divide both sides by -6 y = 2

Find three solutions to 2x + y = 13

x = 1 2(1) + y = 13 2 + y = 13 subtract 2 from both sides y = 11 ordered pair (1, 11) x = 4 2(4) + y = 13 8 + y = 13 subtract 8 from both sides y = 5 ordered pair (4, 5) x = -7 2(-7) + y = 13 -14 + y = 13 add 14 to both sides y = 27 ordered pair (-7, 27)

Which of the following is the CORRECT point-slope form of the equation for a line with slope m = 3 which passes through the point (-2, -1)? a. y + 2 = 3(x + 1) b. y + 1 = 3(x + 2) c. y - 1 = 3(x - 2) d. y - 2 = 3(x - 1)

y + 1 = 3(x + 2)

How would y = 2x - 3 be written in point-slope form? a. y + 3 = 2(x - 0) b. 2x - y = 3 c. y + 2 = 0(x - 3) d. y + 0 = 2(x + 3)

y + 3 = 2(x - 0)

Which of the following is the point-slope form of a linear equation? a. y = mx + b b. y - y1 = m (x - x1) c. Ax + By = C d. m = y2 - y1 / x2 - x1

y - y1 = m (x - x1)

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

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