Using Slope Linear Equations in Two Variables CHAT Pre-Calculus Section. The siplest atheatical odel for relating two variables is linear equation in two variables. It is called a linear equation because its graph is a line. The Slope-Intercept For of the Equation of a Line The graph of the equation b is a line whose slope is and whose -intercept is (0, b). (If ou let = 0, ou get = b, which eans the -intercept of the line is (0, b).) The value of tells us the slope of the line. The slope is the steepness of the line. It is the ratio of change in to the change in. The larger the absolute value of the slope, the steeper the line.
Section. A line with a positive slope rises fro left to right. A line with a negative slope falls fro left to right. A horizontal line has no slope. A vertical line has an undefined slope. Deterining the Slope of a Line Definition of Slope The slope of the nonvertical line through (, ) and, ) is ( where. rise run run rise rise run
Section. If we do not know the slope, we can find it using an two points. Subtracting the s will give us the rise, and subtracting the s will give us the run. (, ) rise run (, ) (, ) *It does not atter which point ou start with, as long as ou use the sae point for the first nubers of the nuerator and denoinator. Eaple: Find the slope of the line containing the points (4, ) and (, -5). Solution: 5 4 8 4
Eaple: Find the slope of the line containing the points (, -9) and (6, -9). CHAT Pre-Calculus Section. Solution: 9 ( 9) 6 0 4 0 This line is horizontal. Its equation is = -9. Eaple: Find the slope of the line containing the points (5, 4) and (5, -8). Solution: 8 4 5 5 0 undefined This line is vertical. Its equation is = 5. Graphing Linear Equations To graph a linear equation, first graph the -intercept and then use the slope to find another point. Draw the line through these two points. 4
Section. Eaple: Sketch the graph of. Solution: Plot the -intercept (0, ). Fro this point go up and to the right units. (The rise is and the run is.) Draw the line through these two points. Note: The graphing calculator does not put arrows on the end of an lines, curves, aes, or asptotes. These lines do, however, go on forever, and when drawing the b hand, should have arrows drawn to show this. 5
Eaple: Sketch the graph of. CHAT Pre-Calculus Section. Solution: Plot the -intercept (0, -). Since the slope is -, think of it as. This eans the rise is - and the run is. To rise -, we actuall fall units. So, fro the point (0, -), go down units and to the right unit. Draw the line through these two points. Writing Linear Equations in Two Variables There are fors of equations that are soeties used to find the equations of lines, depending on the inforation given. The are the point-slope for and the two-point for. 6
Section. Point-Slope For of the Equation of a Line The equation of the line with slope passing through the point, ) is ( ( ) Two-Point For of the Equation of a Line The equation of the line passing through the points, ) and, ) is ( ( ( ) An alternative wa of finding the equation of a line uses onl the slope-intercept for. Eaple: Find the equation of the line with slope 4 that passes through the point (-6, ). Solution: We start with the equation b. We know that our equation ust be 4 b. Since the line passes through the point (-6, ), then this point ust work in the equation. Plug in the point and solve for b. 7
Section. Plug in the point (-6, ) and solve for b. 4 b 4( 6) 4 b b 6 b Replace the value of b into the equation to get 4 6 Eaple: Find the equation of the line that passes through the points (-, ) and (4, 5). Solution: Start with the equation b. We alwas need the slope, so find the slope using the points. rise 5 run 4 6 Use either one of the points in the equation b to find b. 8
Section. Plug in the point (4, 5) and solve for b. b 5 (4) b 5 b b The equation is Eaple: Find the equation of the line that passes through the points (5, ) and (-, ). Solution: Start with the equation b. We alwas need the slope, so find the slope using the points. rise run 5 6 9
Use either one of the points in the equation b to find b. We will use (5, ). CHAT Pre-Calculus Section. b (5) b 5 b Multipl through b to clear the fractions. 5 8 b b 8 b The equation for the line ust be 8. Equations of Lines. General for: A B C 0. Vertical line: a. Horizontal line: b 4. Slope-intercept for: b 5. Point-slope for: ( ) 6. Two-Point for: ( ) 0
Section. Parallel and Perpendicular Lines The slopes of nonvertical lines can be used to deterine whether the lines are parallel or perpendicular. Parallel and Perpendicular Lines Two distinct nonvertical lines are parallel if and onl if their slpes are equal. That is,. Two nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of each other. That is,. (Their product = -) Eaple: Are the lines 6 and 7 parallel or perpendicular? Solution: parallel Eaple: What is the slope of a line perpendicular to the line 6? Solution: -/
Section. Eaple: Find the equation of the line that passes through the point (-, ) and is parallel to the line given b Solution: First put the equation in slope-intercept for to deterine the slope. The line through the point (-, ) ust have the sae slope. Find b. b ( ) b b b The line is.
Section. Eaple: Find the equation of the line perpendicular to the line and passes through the point (4, 9). Give the answer in general for. Solution: The slope ust be the negative reciprocal of, which is. Use this slope and the point to find b. b 9 (4) b 9 6 b b The equation is. Put this in general for. 6 6 0
Section. Application In real-life probles, the slope of a line can be interpreted as either a ratio or a rate. If the -ais and -ais have the sae unit of easure, then the slope has no units and is a ratio. If the -ais and -ais have different units of easure, then the slope is a rate or rate of change. Eaple: A boat rap is built so that the rap rises 4 inches over a horizontal length of 0 feet. What is the slope of the rap? Solution: Since we can change the inches into feet so that the easureents are the sae, we write the slope as rise run ft. 0 ft. 0 (Notice that the easureents cancel each other and we are left with the slope with no unit easure.) 4
Section. Eaple: A copan that anufactures clocks deterines the cost of producing tos is C 5 00 Describe the practical significance of the -intercept and the slope of this line. Solution: Look at the graph $5 $0 Cost $5 $0 $05 $00 $95 $90 4 5 6 7 Nuber of units The -intercept (0, 00) tells us that even if no units are produced, the cost is $00. This fied cost could be things like rent or achines. The slope tells us the cost of producing each unit. 5
Section. If the cost of depreciation is the sae aount ever ear, then we call this linear or straight-line depreciation. Eaple: A copan purchases a $0,000 achine. In 4 ears the achine will be worth $0,000. Write a linear equation that relates the value V of the achine after t ear. Solution: Think of this in ters of points. At 0 ears, the value of the achine is $0,000, so we have the point (0, 0,000). After 4 ears the value is $0,000, so the point is (4, 0,000). Find the slope between these points. 0,000 0,000 0 4 500 Find the equation: b V 500t b 6
Section. Put in one of the points to find b. V 500t b 0,000 500(0) b 0,000 Therefore the equation ust be b V 500t 0,000 Intercept For of an Equation If an equation is of the for, a b then the -intercept is a and the -intercept is b. Eaple: Find the intercepts of 5 Solution: The -intercept is (, 0) and the -intercept is (0, -5). Note: If we have to find the standard for we need to clear the fractions b ultipling through b 5. 7