PRIMARY CONTENT MODULE Algebra - Linear …

1 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-37/H-37 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONWhat does the number m in y = mx + bmeasure?To find out, suppose (x1, y1) and (x2, y2) are twopoints on the graph of y = mx + y1 = mx1 + b and y2 = mx2 + Algebra to simplify y2 y1x2 x1 And give a geometric this! PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-38 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONA nswer:y2 y1x2 x1=mx2+b() mx1+b()x2 x1=mx2 mx1+b bx2 x1=mx2 mx1x2 x1=m(x2 x1)x2 x1 distributive property=mNo matter which points (x1,y1) and (x2, y2) arechosen, m = y2 y1x2 what does this mean? PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-39 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONM eaning of m = y2 y1x2 x1 in y = mx + bm = y2 y1x2 x1 is the rise ( y2 y1) over the run ( x2 x1) andm is called the slope.

2 (x2, y2)y2 y1x2 x1(x1, y1) PRIMARY CONTENT MODULEA lgebra I - Linear equations & InequalitiesT-40/H-40 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeFind the slope, m, of the line whose graphcontains the points (1,2) and (2, 7). PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-41 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING equations olutionm = y2 y1x2 x1 = 7 22 1m = 51m = 5 The rise over the run, or slope, of the line whosegraph includes the points (1,2) and (2,7) is CONTENT MODULEA lgebra - Linear equations & InequalitiesT-42 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONWhat does it mean if the slope, m, is negative iny = mx + b?The negative slope means that y decreases asx some examples. x2 x1(x2,y2)y2 y1(x1,y1)m = y2 y1x2 x1 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-43 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONxy = -2xy = -2x + 2y = -2x 20-2 0 = 0-2 0 + 2 = 2-2 0 2 = -21-2 1 = -2-2 1 + 2 = 0-2 1 2 = -4y = 2xy = 2x 2y = 2x + 2 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-44/H-44 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONDEFINITIONSD efinition 1In the equation y = mx + b for a straight line, thenumber m is called the slope of the 2In the equation y = mx + b for a straight line, thenumber b is called the y-intercept of the CONTENT MODULEA lgebra - Linear equations & InequalitiesT-45 1999, CISC.

3 Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONM eaning of the y-intercept, b, iny = mx + bLet x = 0, then y = m 0 + b,so y = number b is the coordinate onthe y-axis where the graph crosses PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-46 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xample:y = 2x + 3 What is the coordinate on the y-axis where thegraph of y = 2x + 3 crosses y-axis?Answer: 33 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-47 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONThe Framework .. the following idea must be clearlyunderstood before the student can progressfurther:A point lies on a line given by, forexample, the equation y = 7x + 3, ifand only if the coordinates of thatpoint (a, b) satisfy the equation whenx is replaced with a and y is replacedby b. (page 159)Review this statement with the people at yourtable and discuss how you would present this tostudents in your CONTENT MODULEA lgebra - Linear equations & InequalitiesH-48 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONV erify whether the point (1,10) lies on the liney = 7x + CONTENT MODULEA lgebra - Linear equations & InequalitiesT-48 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONV erify whether the point (1,10) lies on the liney = 7x + : If a point lies on the line, its x and ycoordinates must satisfy the x = 1 and y = 10 in the equationy = 7x + 3, we have 10 = 7 1 + 310 = 10 which is true, therefore the point (1,10)lies on the line y = 7x + CONTENT MODULEA lgebra - Linear equations & InequalitiesT-49/H-49 1999, CISC.

4 Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeTell which of the lines this point (2,5) lies on:1. y = 2x + 12. y = 12x + 43. y = 3x + 14. y = 3x + 15. y = 4x + 13 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-50 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xampleSuppose we know that the graph of y = 2x + bcontains the point (1, 2).What must the y-intercept be?Answer: Substitute x = 1 and y = 2 iny = 2x + b, and then solve for = 2 1 + b2 = 2 + b4 = b b = 4 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-51/H-51 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeFind b for the given lines and points on y = 3x + b;(2,7)2. y = 5x + b;( 1, 3)3. y = 12x + b;(4,5) PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-52/H-52 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONG raph y = 3x + 1 by plotting two points andconnecting with a straight CONTENT MODULEA lgebra - Linear equations & InequalitiesT-53/H-53 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xample: y = 2x 5.

5 Use the properties of they-intercept and slope to draw a CONTENT MODULEA lgebra - Linear equations & InequalitiesT-54 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING equations olution:Use b. In the equation y = 2x 5, the y-intercept, b, is 5. This means the line crossesthe y-axis at 5. What is the x coordinate forthis point?The coordinates of one point on the line are(0, 5), but we need two points to graph a ll use the slope to locate a second the equation, we see that m = 2 = 21. Thistells us the rise over the run . We will moveover 1 and up 2 from our first point. The newpoint is (1, 3). rise of 2 run of 1 Verify that (1, 3)satisfies the CONTENT MODULEA lgebra - Linear equations & InequalitiesT-55 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING equations tandard 7 Algebra I, Grade 8 StandardsStudents verify that a point lies on a line givenan equation of a line.

6 Students are able toderive Linear equations using the at the Framework and see how this relatesto the Algebra and function standards for CONTENT MODULEA lgebra - Linear equations & InequalitiesT-56 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIOND etermine the equation of the line that passesthrough the points (1, 3) and (3, 7).Slope = m = y2 y1x2 x1 Step 1: Use the formula above to determine = 7 33 1=42=2 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-57 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONW riting an equation of a line continued:Step 2: Use the formula y = mx + b todetermine the y-intercept, x and y in the formula with thecoordinates of one of the given points, andreplace m with the calculated value, (2).y = mx + bIf we use (1,3) and m = 2, we have3 = 2 1 + b3 = 2 + b1 = b or b = 1If we use the other point (3,7) and m = 2, willwe obtain the same solution for b?

7 7 = 2 3 + b7 = 6 + b1 = b or b = 1So, substituting m = 2 and b = 1 into y = mx + bthe equation of the line is y = 2x + 1 or y = 2x + CONTENT MODULEA lgebra - Linear equations & InequalitiesT-58/H-58 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONG uided PracticeFind the equation of the line whose graphcontains the points (1, 2) and (6,5).The answer will look likey = mx + 1: Find mStep 2: Find bStep 3: Write the equation of the line by writingyour answers from Steps 1 and 2 for m and b inthe equation y = mx + this! PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-59 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING equations olution:Find the equation of the line whose graphcontains the points (1, 2) and (6,5).Step 1: m = y2 y1x2 x1=5 ( 2)6 1=75 Step 2: y = 75x + bSubstitute x = 1 and y = 2 into the equationabove. 2 = 75(1) + b 2 = 75 + b 2 75 = bb = 175 Step 3: y = 75x 175 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-60/H-60 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP racticeFind the equation of the line containing thegiven points:1.

8 (1,4) and (2,7)Step 1:Step 2:Step 3:2. (3,2) and ( 3,4)Step 1:Step 2:Step 3: PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-61 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP oint-Slope FormulaThe equation of the line of slope, m, whosegraph contains the point (x1, y1) isy y1 = m(x x1)Example: Find the equation of the line whosegraph contains the point (2,3) and whose slopeis 3=4(x 2)y 3=4x 8y=4x 5 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-62/H-62 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONP ractice with point-slope formulay y1 = m(x x1)1. Find the equation of the line with a slope of 2and containing the point (5,7)2. Find the equation of the line through (2,7)and (1,3). (Hint: Find m first.) PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-63/H-63 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONH orizontal LinesIf m = 0, then the equation y = mx + b becomesy = b and is the equation of a horizontal : y = 5On the same pair of axes, graph the linesy = 2 and y = CONTENT MODULEA lgebra - Linear equations & InequalitiesT-64 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONWhat about vertical lines?

9 A vertical line consists of all points of the form(x,y) where x = a means x = a constant and y can take : x = 3 What about the slope of a vertical line? Let suse two points on the line x = 3, namely (3,5)and (3,8), then m = 8 53 3=30. Division by 0 isundefined. The slope of a vertical line the same pair of axes,graph the lines x = 3 andx = CONTENT MODULEA lgebra - Linear equations & InequalitiesT-65 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING equations tandard Form for Linear EquationsThe equation Ax + By = C is called the generallinear equation. Any equation whose graph is aline can be expressed in this form, whether theline is vertical or CONTENT MODULEA lgebra - Linear equations & InequalitiesT-66 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONAny non-vertical line is the graph of an equationof the form y = mx + b. This may be rewrittenas mx + y = if A = m, B = 1, and C = b, we getAx + By = , the equation y = mx + b may be expressedin the form Ax + By = CONTENT MODULEA lgebra - Linear equations & InequalitiesT-67 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONE xample:Express y = 3x + 4 in the general Linear formAx + By = 3x + 43x + y=3x 3x + 43x + y=0 + 43x + y=4 Here A = 3, B = 1, and C = about vertical lines?

10 PRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-68 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONAny vertical line has an equation of theform x = k where k is a = kcan be rewritten asAx + By = Cwhere A = 1, B = 0, and C = example, x = 2 can be rewritten as1 x + 0 y = CONTENT MODULEA lgebra - Linear equations & InequalitiesT-69 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONThe general Linear equationAx + By = CCan also be expressed in the formy = mx + bprovided B :Ax + By=CBy= Ax + Cy=1B( Ax + C)y= ABx + CBPRIMARY CONTENT MODULEA lgebra - Linear equations & InequalitiesT-70 1999, CISC: Curriculum and Instruction Steering CommitteeThe WINNING EQUATIONA lgebra PracticeRewrite the equation 2x + 3y = 4 in the formy = mx + : 2x + 3y =43y=2x + 4y=13(2x + 4)y=23x + 43 Here m = 23 and b = 43.