Linear Equations Linear Inequalities Drive on The Education Highway

Linear Equations and Graphing 1. Parts of a Coordinate Plane 2. Slope 3. Slope-intercept Form of a Linear Equation 4. Graphing by x- & y-intercepts.

Parts of a coordinate plane.

Click on the correct quadrant numbers. Correct answer = applause. Quadrant IIIIIIIV Quadrant IIIIIIIV Quadrant IIIIIIIV Quadrant IIIIIIIV Lesson Start

Click on the correct axis names. Correct answer = clapping. x-axis y-axis x-axisy-axis Lesson Start

Click on the point for the origin. Correct answer = clapping. x y Lesson Start

Click on point (-3, 5). Correct point = applause. x y Lesson Start

You chose a line segment instead of a point. Go back and try again.

You chose point (5, -3). Each ordered pair is in the form (x, y) -- it follows the alphabet in its internal order. You find the x value first, then you find the y value. Where they meet is the point. Go back and try again.

Click on the correct ordered pair for the black point. -5/4 (4, -5) (-5, 4) -4/5 x y Lesson Start

You did not choose an ordered pair. Go back and try again.

You chose the ordered pair for the pale green point. Remember: x comes before y in the alphabet and in an ordered pair. Go back and try again.

Return to Main Menu. Return to Prior Problem. Continue to Next Lesson.

Slopes 1. What is a slope? 2. Slope formula 3. Types of slopes

What is slope? Slope is the slant of a line. Slope = risechange in y’s runchange in x’s Slope is a fraction/integer. Lesson Start

x y How to determine the slope when the line goes up. 1. Count the number of units up from the right point to the left point Put that number on top of the fraction line. Slope = 6 3, Count the number of units to the right Put that number under the fraction line. 9 Lesson Start

x y How to determine the slope when the line goes down. 1. Count the number of units down from right point to left point Put that number on top of fraction line. Slope = Count the units to the right Put that number under the fraction line. 3 Lesson Start

x y Determine the slope of the line shown. -1/3 3/1 -3/11/3 Lesson Start

The line does not go down. Go back and try again. Lesson Start

The line does not rise 3 units, then run 1 unit to the right. Go back and try again. Lesson Start

x y Determine the slope of the line shown. -2/33/2 -3/22/3 Lesson Start

The line does not go up. Go back and try again. Lesson Start

The line does not rise -2 units, then run 3 units to the right. Go back and try again. Lesson Start

Slope Formula: m = (y 1 - y 2 ) (x 1 - x 2 ) where m = slope and (x 1, y 1 ), (x 2, y 2 ) are points on the line. Lesson Start

Example: Find the slope for a line with points (-3, 4) and (7, -2) on it. 1. Assign values as follows: 2. Substitute them into the formula and solve. m = 4 - (-2) = 6 = (x 1, y 1 ) = (-3, 4) (x 2, y 2 ) = (7, -2) Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1 ) = (5, 6)(2, 9) Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1 ) = (5, 6) 2. (x 2, y 2 ) = (5, 6)(2, 9) Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1 ) = (5, 6) 2. (x 2, y 2 ) = (2, 9) 3. m = Lesson Start

The slope formula is a case of subtraction on top and bottom. Go back and try again. Lesson Start

You have your x’s and y’s upside down. Remember: “Y’s guys are always in the skies!” Go back and try again. Lesson Start

You have your x’s and y’s upside down. You are also adding when you need to subtract. Go back and try again. Lesson Start

Find the slope of the line with points (5, 6) and (2, 9) on it. 1. (x 1, y 1 ) = (5, 6) 2. (x 2, y 2 ) = (2, 9) 3. m = = -3 = Lesson Start

Find the slope of the line with points (7, 5) and (3, -4) on it. m = = = (-4) (-4) = (-4) = Lesson Start

You have your x’s and y’s upside down. Remember: “Y’s guys are always in the skies!” Go back and try again. Lesson Start

It is not 5 - 4, it is 5 - (-4). Go back and try again. Lesson Start

You must start with the y and x from the first point and end with the y and x from the second point. Go back and try again. Lesson Start

Types of Slopes: 1. Positive and Negative Slopes 2. Special Types of Slopes 3. Determining Types of Slopes by Looking at Graphs of Lines 4. Determining Types of Slopes Algebraically. Lesson Start

Positive and Negative Slopes. TypeGraph Algebra PositiveUp left to right.Positive Fraction NegativeDown left to right. Negative Fraction Lesson Start

2 Special Types of Slopes TypeGraphsAlgebra ZeroHorizontal Line 0/a, a  0 UndefinedVertical Line a/0 No Slope Lesson Start

Determining Types of Slopes by Looking at Graphs of Lines Lesson Start

x y Is the slope of the line positive, negative, zero, or undefined? - 0  + Lesson Start

x y Is the slope of the line positive, negative, zero, or undefined? - 0  + Lesson Start

x y Is the slope of the line positive, negative, zero, or undefined? - 0  + Lesson Start

x y Is the slope of the line positive, negative, zero, or undefined? - 0  + Lesson Start

x y Click on the line with the negative slope. Lesson Start

x y Click on the line with the zero slope. Lesson Start

x y Click on the line with the no slope. Lesson Start

x y Click on the line with the positive slope. Lesson Start

Determining Types of Slopes Algebraically. Lesson Start

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (-3, 5) and (-9, -4) +-0  Lesson Start

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (3, 5) and (3, -4) +-0  Lesson Start

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (-3, -5) and (-9, -4) +-0  Lesson Start

Is the slope of the line with the 2 points listed positive, negative, zero, or undefined? Points (-3, -4) and (-9, -4) +-0  Lesson Start

Return to Main Menu. Return to Prior Problem. Continue to Next Lesson.

Slope-intercept Form of a Linear Equation 1. Slope-intercept equation 2. Graphing by slope-intercept 3. Writing slope-intercept equations

Slope-intercept Form: y = mx + b where m = slope and b = y-intercept. Lesson Start

Example: y = -1/2x /2 = m = slope 4 = b = y-intercept Lesson Start

Graphing by Slope-intercept 1. Graphing lines with positive/negative slopes. 2. Graphing lines with zero or undefined/no slopes. Lesson Start

The Slope-intercept Song You make the last number first. It’s either up or down. Make the slope in 2 numbers, Or you look like a clown. Top one’s up or down, And the bottom’s always right. You’d better do it well, Or you’ll get a fright. (Tune: “Hokey-Pokey”) Lesson Start

x y Graph y = -1/2x Last number is 4, so go up 4 on the y- axis from the origin and plot a point Slope is already 2 numbers. Top one is -1, so go down 1 from the point you just plotted. 3. The bottom number is 2, so go 2 units to the right and plot a point. 4. Draw a line through the 2 points you plotted. Lesson Start

x y Graph y = 2x Last number is -3, so go down 3 units from the origin and plot a point The slope is only 1 number so put a 1 under the Go up 2 from the point you just plotted. 4. Go 1 unit to the right and plot a point. 5. Draw a line through the 2 points you plotted. 1 Lesson Start

x y Click on the graph for y = 2/3x - 2 Lesson Start

The slope is not negative. Go back and try again. Lesson Start

You graphed the last number on the x-axis instead of the y-axis. Go back and try again. Lesson Start

Top number is 2, and the bottom is 3, so you do not go up 3 and over 2. Go back and try again. Lesson Start

x y Click on the graph for y = -4x + 3 Lesson Start

The slope is not positive. Go back and try again. Lesson Start

The -4 is not the y-intercept, nor is the 3 the x-intercept. Go back and try again. Lesson Start

The -4 is the slope, not the x-intercept. Go back and try again. Lesson Start

Two Special Graphs: Line with a zero slope And Line with an undefined slope. Lesson Start

Line with a zero slope: y = # (no x) graphs as a horizontal line. “Why, o y, do I look upon the horizon?” Lesson Start

x y Graph the equation y = 2. Lesson Start

Line with an undefined/no slope: x = # (no y) graphs as a vertical line. Lesson Start

x y Graph the equation x = -4. Lesson Start

x y Click on the graph for x = 3. Lesson Start

You chose the graph for x = -3. Go back and try again. Lesson Start

The x = # (no y) line does not graph as a horizontal line. Go back and try again. Lesson Start

x y Click on the graph for y = -3½. Lesson Start

The y = # (no x) line does not graph as a vertical line. Go back and try again. Lesson Start

You chose the graph for y = 3½. Go back and try again. Lesson Start

Writing Slope-intercept Equations: 1. When given a slope and the y-intercept. 2. When given a slope and one point on the line. 3. When given 2 points on the line. m = ¾, b = -1 m = -¼, (8, 3) (3, 7), (5, 12) Lesson Start

Writing a slope-intercept equation when given a slope and the y-intercept. Substitute the slope and the y-intercept for the m and b in the equation. Example: m = ¾, b = -1 y = mx + b Slope-int. equation y = ¾x - 1 The new equation Lesson Start

1. Click on the correct equation for a line with slope = 5/3 and y-intercept = 2. y = 5/3x + 2y = 2x + 5/35/3y = 2xy = -5/3x Click on the correct slope and y-intercept pair for y = 7x - 5. m = 7, b = -5m = -5, b = 7m = -7, b = 5m = 1/7, b = -5 Lesson Start

Writing a slope-intercept equation when given a slope and a point on the line. 1. Substitute the x, y, and m in the slope-intercept form. 2. Solve for b. 3. Substitute the slope and the b in a clean slope-intercept form. Lesson Start

Example: Write the equation of the line with slope = -¼ and point (8, 3). y = mx + b 3 = -¼(8) + b 1. Substitute the slope, x, and y in the equation. 3 = -2 + b +2 5 = b 2. Solve for b. y = -¼x Substitute the slope and b in the equation. Lesson Start

1. Click on the correct substitution for a line with slope = 1/3 and point (5, 9). 9 = 1/3(5) + b5 = 1/3(9) + b9 = 1/3x + 59y = 5x + 1/3 2. Click on the correct equation for a line with slope = -2/3 and point (-6, 4). y = -2/3xY = -2/3x + 4y = -2/3y = -2/3x - 6 Lesson Start

Writing a slope-intercept equation for a line with 2 points given: 1. Find the slope of the line. 2. Use that slope and the first point to find the y-intercept. 3. Substitute the slope and the y-intercept into the equation. Lesson Start

Example: Write and equation for a line with points (3, 7) & (5, 12). 1. Find the slope of the line. m = (y 1 - y 2 ) (x 1 - x 2 ) m = = -5 = Continued on next screen. Lesson Start

Write and equation for a line with points (3, 7) & (5, 12). m = 5/2 y = mx + b 2. Use the slope and the first point to solve for the y-intercept. 7 = (5/2)(3) + b 2(7) = 2(15/2) + 2b 14 = b = 2b  -1/2 = b 2 2 Continued on next screen. Lesson Start

Write and equation for a line with points (3, 7) & (5, 12). m = 5/2, b = -1/2 3. Substitute the slope and the y-intercept for the m and the b in the equation. y = mx + b y = 5/2x - 1/2 Lesson Start

1. Click on the slope for a line with points (-2, 8) and (7, -5) Click on the y-intercept for a line with points (-2, 8) and (7, -5) Lesson Start

3. Click on the correct equation for a line with points (3, 7) and (4, 8). y = x + 4 y = 3x + 7 y = -x + 4 y = 3/4x + 8 Lesson Start

Return to Main Menu. Return to Prior Problem. Continue to Next Lesson.

Graphing by x- and y-intercepts. X-intercept: where the line crosses the x-axis. Y-intercept: where the line crosses the y-axis. x y x-intercept y-intercept

How to graph by x- & y-intercepts: 1. Cover the x term with your index finger and solve the resulting equation for y. 2. Go up or down on the y-axis from the origin that many units and plot a point. 3. Cover the y term with your index finger and solve the resulting equation for x. 4. Go left or right on the x-axis from the origin that many units and plot a point. 5. Draw a line through your points. Lesson Start

x y Graph the line for 3x + 2y = Cover the x term and solve for y. 3x + 2y = 6. y = 3 2. Go up 3 units on the y-axis Cover the y term and solve for x. 3x + 2y = 6. x = 2 4. Go right 2 units on the x-axis Draw a line through the points plotted. Lesson Start

1. Click on the correct intercepts for 3x - 4y = 24. x-int: 8 y-int: -6 x-int: -6 y-int: 8 x-int: 8 y-int: 6 x-int: 6 y-int: 8 Lesson Start

x y 2. Click on the graph of 3x - 6y = 12. Lesson Start

x y 3. Click on the correct equation for the line shown. -6x - 9y = x - 9y = y - 6x = y - 6x = -36 4x + 6y = 36 4x + 6y = 36 6y + 4x = 36 6y + 4x = 36 Lesson Start

Return to Main Menu. Return to Prior Problem. Continue to Next Lesson.

Graphing Linear Inequalities Type of Line Where to Shade Solving Inequalities

How to Determine the Type of Line to Draw Inequality Symbol Type of Line > or <Dotted Line > or <Solid Line

Choose the type of line for the inequality given. 1. y > 3x - 2 a. Solidb. Dotted 2. y > ¼x - 5 a. Solidb. Dotted Lesson Start

Choose the inequality symbol for the line shown. < or > < or > Lesson Start

Choose the inequality symbol for the line shown. < or > < or > Lesson Start

For Positive, Negative, & Zero Slopes For Undefined or No Slopes

Where to Shade for Positive, Negative, and Zero Slopes: The inequality must be in y  mx + b format.  can be: >, >, <, or <. Lesson Start

If the inequality is: Shade y > mx + b or y > mx + b Above the line y < mx + b or y < mx + b Below the line Lesson Start

x y Graph y > x Graph the line y = x Since y >, shade above the line. Lesson Start

x y Graph y < x Graph the line y = x Since y <, shade below the line. Lesson Start

Do you do anything different when the line is dotted rather than solid? Lesson Start

Lesson Start

x y Graph y > x Since y >, shade above the line. 1. Graph the line y = x - 2, but make the line dotted. Lesson Start

x y Graph y < x Graph the line y = x - 2, but make the line dotted. 2. Since y <, shade below the line. Lesson Start

x y Graph y > -½x + 3 Type of line: Solid Dotted Lesson Start

x y Graph y > -½x + 3 Type of line: Solid Dotted Shade ___ the line. AboveBelow Lesson Start

x y Graph y > -½x + 3 Type of line: Solid Dotted Shade ___ the line. AboveBelow Lesson Start

x y Choose the correct inequality for the graph shown. y < 1/3 x + 2 y < 1/3 x + 2 y > 1/3 x + 2 y > 1/3 x + 2 Lesson Start

Where to Shade for Undefined or No Slopes: The inequality must be in x  # (no y) format.  can be: >, >, <, or <. Lesson Start

If the inequality is: Shade To the x > # or x > # Right of the line x < # or x < # Left of the line Lesson Start

x y Graph x > Draw a dotted vertical line at x = Shade to the right of the line. Lesson Start

x y Graph x < Graph the line X = Shade to the left of the line. Lesson Start

x y Graph x > 3. Choose type of line. Solid Dotted Lesson Start

x y Graph x > 3. Choose type of line. Solid Choose where to shade. Left Right Lesson Start

x y Graph x > 3. Choose type of line. Solid Choose where to shade. Right Lesson Start

Solving Inequalities You use the same algebraic methods as solving equations except when you multiply/divide both sides by the same negative number. In that case, you switch the direction of the inequality symbol. Lesson Start

Solve -3x - 2y < x - 2y < 12 +3x -2y < 3x y < -3/2 x - 6> Lesson Start

Choose the correct inequality. 1. 2x + 5y > -10 y > -2/5 x - 2 y < -2/5 x - 2 y > 2/5 x + 2 y < 2/5 x x - 2y > 10 y > -2/3 x - 5 y < -2/3 x - 5 y > 2/3 x - 5 y < 2/3 x - 5 Lesson Start