Review Linear Equations and Graphs

Linear Equations in Two Variables A linear equation in two variables is an equation that can be written in the standard form Ax + By = C, –where A, B, and C are constants (A and B not both 0), and –x and y are variables. –Linear equations graph lines. Anything not straight is called a curve. A solution of an equation in two variables is an ordered pair of real numbers that satisfy the equation. –For example, (4,3) is a solution of 3x - 2y = 6. The solution set of an equation in two variables is the set of all solutions of the equation. The graph of an equation is the graph of its solution set.

Linear Equations A linear equation can be written in the forms: where A, B, and C are integers and x and y are variables. (whole numbers) STANDARD FORM: 1. 2.A linear equation graphs a straight line. SLOPE-INTERCEPT FORM: where m is the slope and b is the y-intercept. POINT-SLOPE FORM: where m is the slope and (x 1,y 1 ) is a point.

Graphing Linear Equations

What is Intercept in Math? Y X An Axis Intercept is the point where a line crosses the x or y axis

Intercepts of a Line y-intercept: where the graph crosses the y-axis. The coordinates are (0, b). x-intercept: where the graph crosses the x-axis. The coordinates are (a, 0). To find the x-intercept, let y = 0 and solve for x. To find the y-intercept, let x = 0 and solve for y.

Introduction Linear Graphs When to use the x and y intercepts to graph linear equation  Given the equation in Standard Form Ax + By = C

Introduction Linear Graphs Finding Intercepts: In the equation of a line, let y = 0 to find the “x-intercept” and let x = 0 to find the “y-intercept”. Note: A linear equation with both x and y variables will have both x- and y-intercepts. Example: Find the intercepts and draw the graph of 2x –y = 4 x-intercept: Let y = 0 : 2x –0 = 4 2x = 4 x = 2 y-intercept: Let x = 0 : 2(0) – y = 4 -y = 4 y = -4 x-intercept is (2,0) y-intercept is (0,-4)

Introduction Linear Graphs Finding Intercepts: In the equation of a line, let y = 0 to find the “x-intercept” and let x = 0 to find the “y-intercept”. Example: Find the intercepts and draw the graph of 4x –y = -3 x-intercept: Let y = 0 : 4x –0 = -3 4x = -3 x = -3/4 y-intercept: Let x = 0 : 4(0) – y = -3 -y = -3 y = 3

Example: 2x + 3y = 5 x-intercept 2x + 3(0) = 5 2x + 0 = 5 2x = 5 y-intercept 2(0) + 3y = y = 5 3y = 5

Example: 4x - y = 6 x-intercept 4x - (0) = 6 4x - 0 = 6 4x = 6 y-intercept 4(0) - y = y = 6 -y = 6

Example: 4x + 2y = 3 x-intercept 4x + 2(0) = 3 4x + 0 = 3 4x = 3 y-intercept 4(0) + 2y = y = 3 2y = 3 Jeff Bivin -- LZHS

Example: 3x - 2y = 7 x-intercept 3x - 2(0) = 7 3x - 0 = 7 3x = 7 y-intercept 3(0) - 2y = y = 7 -2y = 7 Jeff Bivin -- LZHS

Introduction Linear Graphs Graphing a line that passes through the origin: Some lines have both the x- and y-intercepts at the origin. Note: An equation of the form Ax + By = 0 will always pass through the origin. Find a multiple of the coefficients of x and y and use that value to find a second ordered pair that satisfies the equation. Example: A) Graph x + 2y = 0

The Slope of a Line Finding the slope of a line given an equation of the line: The slope can be found by solving the equation such that y is solved for on the left side of the equal sign. This is called the slope-intercept form of a line. The slope is the coefficient of x and the other term is the y- intercept. The slope-intercept form is y = mx + b Example : Find the slope of the line given 3x – 4y = 12

The Slope of a Line Finding the slope of a line given an equation of the line: Example : Find the slope of the line given y + 3 = 0 y = 0x - 3  The slope is 0 Example : Find the slope of the line given x + 6 = 0 Since it is not possible to solve for y, the slope is “Undefined” Note: Being undefined should not be described as “no slope” Example : Find the slope of the line given 3x + 4y = 9

Introduction Linear Graphs Recognizing equations of vertical and horizontal lines: An equation with only the variable x will always intersect the x-axis and thus will be vertical. An equation with only the variable y will always intersect the y-axis and thus will be horizontal. Example: A) Draw the graph of y = 3 B) Draw the graph of x + 2 = 0x = -2 A) B)

Graphing Horizontal Lines This line has a y value of 4 for any x-value. It’s equation is y = 4 (meaning y always equals 4)

Graphing Vertical Lines This line has a x value of 1 for any y-value. It’s equation is x = 1 (meaning x always equals 1)

The Equation of a Vertical Line is X=Constant x = 1

The Equation of a Horizontal Line is Y=Constant y = 3

Graph the following lines Y = -4 Y = 2 X = 5 X = -5 X = 0 Y = 0

Answers x = 5 x = -5

Answers y = -4 y = 2

Answers x = 0 y = 0 Horizontal and Vertical lines are always perpendicular to each other

Slope of a Line

SLOPE = Slope is a measure of STEEPNESS

The Symbol for SLOPE = m Think of m for Mountain

Slope of a Line Slope of a line: rise run Note: The slope of a line is the SAME everywhere on the line!!! You may use any two points on the line to find the slope.

The Slope of a Line Finding the slope of a line given two points on the line: The slope of the line through two distinct points (x 1, y 1 ) and (x 2, y 2 ) is: Note: Be careful to subtract the y-values and the x-values in the same order. Correct Incorrect

The Slope of a Line Finding the slope of a line given two points on the line: Example : Find the slope of the line through the points (2,-1) and (-5,3)

m=SLOPE = (3,2) (6,4) (0,0) x1y1x1y1 x2y2x2y2

(x 2,y 2 )(6,4) (x 1,y 1 )(3,2) Switch points and calculate slope Make (3,2) (x 2,y 2 ) & (6,4) (x 1,y 1 ) (x 1,y 1 )(6,4) (x 2,y 2 )(3,2)

Recalculation with points switched (x 1,y 1 )(6,4) (x 2,y 2 )(3,2) Same slope as before

It doesn’t matter what 2 points you choose on a line the slope must come out the same

Keeping Track of Signs When Finding The Slope Between 2 Points Be Neat & Careful Use (PARENTHASES) Double Check Your Work as you Go Follow 3 Steps

3 Steps for finding the Slope of a line between 2 Points (3,4)&(-2,6) 1st Step: Write x 1,y 1,x 2,y 2 over numbers 2nd Step: Write Formula and Substitute x 1,x 2,y 1,y 2 values. 3rd Step: Calculate & Simplify (3,4) & (-2,6) x 1 y 1 x 2 y 2

Find the Slopes of Lines containing these 2 Points 1. (1,7) & (5,2) 5. (3,6) & (5,-5) 3. (-3,-1) & (-5,-9) 6. (1,-4) & (5,9) 4. (4,-2) & (-5,4) 2. (3,5) & (-2,-8)

1. (1,7) & (5,2) 5. (3,6) & (5,-5) 3. (-3,-1) & (-5,-9) 6. (1,-4) & (5,9) 4. (4,-2) & (-5,4) 2. (3,5) & (-2,-8) ANSWERS

Solve for y if (9,y) & (-6,3) & m=2/3

Review Finding the Slopes of Lines Given 2 Points 1st Step: Write x 1,x 2,y 1,y 2 over numbers 2nd Step: Write Formula and Substitute x 1,x 2,y 1,y 2 values. 3rd Step: Calculate & Simplify NOTE: Be Neat, Careful, and Precise and Check your work as you go..

ZERO Slope Horizontal Positive Slope Is Up the Hill Negative Slope Is Down the Hill NO Slope Vertical Drop

ZERO Slope Horizontal NO Slope Vertical Drop

Interpreting Slope Lines that increase from left to right have a Positive slope. Lines that decrease from left to right have a Negative slope. x y x y

Interpreting Slope Lines that are horizontal have a slope of Zero. Lines that are vertical have an Undefined slope. x y x y

The Slope of a Line Graph a line given its slope and a point on the line: Locate the first point, then use the slope to find a second point. Note: Graphing a line requires a minimum of two points. From the first point, move a positive or negative change in y as indicated by the value of the slope, then move a positive value of x. Example : Graph the line given slope = passing through (-1,4) Note: change in y is +2

The Slope of a Line Graph a line given its slope and a point on the line: Locate the first point, then use the slope to find a second point. Example : Graph the line given slope = -4 passing through (3,1) Note: A positive slope indicates the line moves up from L to R A negative slope indicates the line moves down from L to R

The Slope of a Line Using slope to determine whether two lines are parallel, perpendicular, or neither: Two non-vertical lines having the same slope are parallel. Two non-vertical lines whose slopes are negative reciprocals are perpendicular. Example : Is the line through (-1,2) and (3,5) parallel to the line through (4,7) and (8,10)?

The Slope of a Line Using slope to determine whether two lines are parallel, perpendicular, or neither: Two non-vertical lines having the same slope are parallel. Two non-vertical lines whose slopes are negative reciprocals are perpendicular. Example : Are the lines 3x + 5y = 6 and 5x - 3y = 2 parallel, perpendicular, or neither?

Linear Equations in Two Variables Writing an equation of a line given its slope and y- intercept. The slope can be found by solving the equation such that y is solved for on the left side of the equal sign. This is called the slope-intercept form of a line. The slope is the coefficient of x and the other term is the y- intercept. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. Example: Find an equation of the line with slope 2 and y-intercept (0,-3) Since m = 2 and b = -3, y = 2x - 3

Equations of a Line There are 3 Forms of Line Equations Standard Form: Ax+By=C Slope Intercept Form: y=mx+b Point-Slope Form y-y 1 =m(x-x 1 ) All 3 describe the line completely but are used for different purposes. You can convert from one form to another.

Converting from Standard Form:ax+by=c to Slope Intercept Form Slope Intercept Form:y=mx+b JUST SOLVE FOR Y

Converting from Slope Intercept Form: y=mx+b to Standard Form Standard Form: Ax+By+C=0 Make the equation equal to 0

Graphing linear Equations

Linear Equations in Two Variables Graphing a line using its slope and y-intercept: Example: Graph the line using the slope and y-intercept: y = 3x - 6 Since b = -6, one point on the line is (0,-6). Locate the point and use the slope (m = ) to locate a second point. (0+1,-6+3)=(1,-3)

y = 2x + 1 Rise Run 2 1 slope y-intercept b = 1 m = 2 y = 2x + 1

y = -3x + 2 Rise Run -3 1 slope y-intercept b = 2 m = -3 y = -3x + 2

Rise Run 2 3 slope y-intercept b = -1

Rise Run 2 slope y-intercept b = 3

Review Steps of Graphing from the Slope Intercept Equation 1.Make sure equation is in y=mx+b form 2.Plot b(y-intercept) on graph (0,b) 3.From b, Rise and Run according to the slope to plot 2nd point. 4.Check sign of slope visually

Point-Slope Form Cross-multiply and substitute the more general x for x 2 where m is the slope and (x 1, y 1 ) is a given point. It is derived from the definition of the slope of a line: The point-slope form of the equation of a line is

Linear Equations in Two Variables Writing an equation of a line given its slope and a point on the line: The “Point-Slope” form of the equation of a line with slope m and passing through the point (x 1,y 1 ) is: y - y 1 = m(x - x 1 ) Where m is the given slope and x 1 and y 1 are the respective values of the given point. Example: Find an equation of a line with slope and a given point (3,-4)

Find the Equation of a Line (Given Pt. & Slope) Using the Pt.-Slope Eq. Given a point (2,5) & m=5 Write the Equation 1.Write Pt.-Slope Equation 2.2. Plug-in (x,y) & m values 3.Solve for y

Linear Equations in Two Variables Writing an equation of a line given two points on the line: The standard form for a line was defined as Ax+By=C. Example: Find an equation of a line passing through the points (-2,6) and (1,4). Write the answer in standard form. Step 1: Find the slope: Step 2: Use the point-slope method:

Parallel Lines Have the Same Slope (0,0)

Perpendicular Lines Have Neg. Reciprocal Slopes (0,0)

Linear Equations in Two Variables Finding equations of Parallel or Perpendicular lines: If parallel lines are required, the slopes are identical. If perpendicular lines are required, use slopes that are negative reciprocals of each other. Example: Find an equation of a line passing through the point (-8,3) and parallel to 2x - 3y = 10. Step 1: Find the slope Step 2: Use the point-slope method of the given line

Linear Equations in Two Variables Finding equations of Parallel or Perpendicular lines: Example: Find an equation of a line passing through the point (-8,3) and perpendicular to 2x - 3y = 10. Step 1: Find the slope Step 3: Use the point-slope method of the given line Step 2: Take the negative reciprocal of the slope found

Summary of Linear Graphs