Pre ctivity Ordered Pairs, Intercepts, and Slopes …

1 Pre-ActivityPrePArAtionSection introduced validating Ordered pair solutions to equations in two variables. This activity shows how to find Ordered pair solutions to an equation in two variables by choosing one variable value and applying the general methodology for solving equations to find the value of the other variable. The following equations are a small sample of the widespread applications of finding Ordered pair solutions to equations in two variables. FoodYou have $20 to buy the chips and salsa for a large party. If chips cost $3 and a jar of salsa costs $2, how many jars of salsa can you buy if you buy 4 bags of chips?$3C + $2S = $20$3(4) + $2S = $20 SportsIn football, field goals score 3 points each and touchdowns score 7 points (with the extra point). If a team scores 2 field goals, how many touchdowns does it score if the total score is 48 points?

2 3F + 7T = 483(2) + 7T = 48 FinanceMoney is divided between two accounts. One is low risk and returns 6% interest. The other is higher risk and returns 11%. If you put $4000 in the low risk account, how much should you put in the high risk account to earn a total of $900 in interest? + = $ + (4000) = $900 ScienceConverting between Fahrenheit and Celsius is common in Chemistry and Physics. The conversion formula is:CF=-9532()Convert F to C:C=-95986 32( .) Find Ordered pair solutions Find the x- and y-intercept of a line from its equation Find the slope of a line from its equation Calculate the slope of an equation, given any two Ordered pairsOrdered Pairs, intercepts , and SlopesSection terms to LeArnslopeslope-intercept formstandard form x-intercepty-interceptPreviously usedcoefficientequationintegerLeArning objectivesterminoLogylinear equationordered pair solutionreal numbers Chapter Graph ngbuiLding mAthemAticAL LAnguAgeLinear Equations Equations in two variables are classified by the exponent (power) of the variables.

3 First degree equations, that is, equations whose variables are raised to the first power, are linear equations. That means that when these equations are graphed in the Rectangular Coordinate System, their graphs make a straight line. Linear equations fit a pattern that looks like Ax + By = C, where x and y are the variables and A, B, and C are real number constants with A and B not both equal to zero. Ax + By = C is called the standard form equation of a example, the equation 2x + 3y = 10 is a linear equation with infinitely many solutions; (2, 2), (5, 0), and ( 1, 4) are just a few of them. You can see from the graph of this line (the figure at right) that solutions to a linear equation are points on the graph of that equation. x and y-InterceptsThere are two Ordered pair solutions that convey extra information and that have special names, the x- and y- intercepts .

4 The x-intercept is the point at which the graph of the line crosses the x-axis. As you can see in the figure at right, that point is (5, 0). It is important to note that the y-coordinate of this point is zero. Similarly, the y-intercept is the point at which the graph of the line crosses the y-axis. The x-coordinate at this point is zero. Though we have not plotted this point on the graph of the line, you can see that it does exist at the point where the line crosses the y-axis. The x and y- intercepts allow us to find two Ordered pair solutions with minimal effort. The point at which the line crosses the x-axis is the x-intercept at (x, 0). The point at which the line crosses the y-axis is the y-intercept at (0, y).y = mx + b: The slope -Intercept Form of the Equation of a LineThere is a special form of the equation for a line that allows us to read information directly from it.

5 The slope -intercept form for the equation of a line looks like y = mx + b. From the slope -intercept form of an equation, we can determine two important pieces of information by inspection just by reading the equation. The first is m, the coefficient of x. That value has special meaning for the graph of the line. It is the slope of the line, or how steep or inclined the line is. The value b in the slope -intercept form indicates the second piece of information, the y-intercept, which is the point (0, b).yx(5, 0)(-1, 4)(2, 2)y-interceptx-intercepty = m x + by-interceptslope Sect on . Ordered Pa rs, intercepts , and Slopesyx(-1, 0)(0, 2)Linear Equation: 2x + y = 2 or y = 2x + 2x-intercept: ( 1, 0) y-intercept: (0, 2) slope : 2yx(0, 1)(1, 0)Linear Equation: x + y = 1 or y = x + 1x-intercept: (1, 0) y-intercept: (0, 1) slope : 1 Note that when the slope is a positive value, the line extends up, left to right.

6 When the slope is a negative value, the slope extends down, left to right. Calculating the slope of a LineIt is possible to calculate the slope of a line, if you know two points on that line ( , two solutions to the equation of the line). It is not difficult, but does require that you understand what the slope of a line actually means. Think of a flight of stairs each step consists of a riser (the vertical part) and the tread (the horizontal part). The flight of stairs consists of rises and runs. The slope of the stairs is a ratio of the rise or vertical change to the run or horizontal have already learned to calculate the length of both vertical and horizontal line segments; the slope is simply the vertical length divided by the horizontal s look at the slope of a line on the coordinate plane. (The equation for this line is: y = 2x 5.)

7 In the figure at right, we can see that the rise is the vertical line segment that extends from 1 to 3 on the y-axis. We can calculate the length of this line as |y2 y1| or |3 ( 1)| = 4. The run is measured on the horizontal line segment that extends from 2 to 4 on the x-axis. We can calculate the length of this line as |x2 x2| or |4 2| = ratio of the rise to the run is ()()yyxx2121--, in this case 42 or 2. This is the slope of the line. The slope Formula allows us to perform these calculations in one neat step. RUNhorizontal(x-direction)RISE vertical(y-direction)yx(2, -1)(4, 3)(2, 3) slope FormulaGiven the coordinates of any two points (x1, y1) and (x2, y2), the slope of their line is:myyxx=--()()2121 0 Chapter Graph ngtechniQues2 Find the x- and y- intercepts for a Linear Equation1. To find an x-intercept, let y = 0 and solve for To find a y-intercept, let x = 0 and solve for Validate by substitution.

8 Example 2: Find the x and y- intercepts for the equation 2x + 3y = 10x-intercept: let y = 0 and solve for x:2 +3(0)=102 =102 10=22=5 xxxxThe x -intercept is (5, 0).Validate: 2(5) + 3(0) ?= 10 10 = 10 y-intercept: let x = 0 and solve for y:2(0)+3 =103 =103 10=3310= 3yyyy10 The y -intercept is (0, ).3 Validate: 2(0) + 3(103) ?= 10 10 = 10 1 Find Ordered Pair Solutions to a Linear Equation1. Choose any value for x (or y).2. Solve for y (or x).3. Validate by substituting the Ordered pair back into the equation. Example 1: Find an Ordered pair solution for the equation 2x + 3y = 10 Choose a value for x: Let x = 42 +3 =102( 4)+3 =10 8+3 =10+8 8+3 =10+83 =18318=33= 6xyyyyyyyAn Ordered pair solution is: ( 4, 6)Validate: 2( 4) + 3(6) ?= 10 8 + 18 ?= 10 10 = 10 or y: Let y = 22 +3 =102 +3( 2)=102 6=102 6+6=10+62 =16216=22= 8xyxxxxxxAn Ordered pair solution is: (8, 2)Validate: 2(8) + 3( 2) ?

9 = 10 16 6 ?= 10 10 = 10 Sect on . Ordered Pa rs, intercepts , and Slopes3 Find the slope of a Linear Equation1. Find two Ordered pair solutions to the equation (use Technique 1).2. Substitute the Ordered pair values into the slope Formula to determine the Validate by finding a third Ordered pair solution to the equation and using this point (together with one of the previous two Ordered pair solutions) to recalculate the slope . Example 3: Find the slope for the equation 2x + 3y = the values previously calculated from Example 1:( 4, 6) and (8, 2)2121( )m=( )( 2 6)m=(8 ( 4))( 8)m=(12) y yxx2= 3slope Validate:Choose a value for x: Let x = 2An Ordered pair solution is: (2, 2).2 +3 =102(2)+3 =104+3 =104 4+3 =10 43 = 6= 2xyyyyyyLet (2, 2) be P1 and use ( 4, 6) as P2. Substitute the values into the slope Formula: 2121( )m=( )(6 2)m=( 4 2)(4)m=( 6)2= 3slope y yxxWe can rearrange the original equation (2x + 3y = 10) into slope -intercept form and visually check the slope and y-intercept:231022310 23210332310323103xyxxyxyxyxyx+=-++=-=-+= -+=-+y = m x + b210= + 33yx Chapter Graph ngmodeLsModel 1 Find two Ordered pair solutions for the equation x 2y = 8:Pick a value for xPick a value for yLet x = 4 Let y = 1 Solve for ySolve for xxyyyyyy-=-=--=--=--=-=-284 284 4 28 42422422 Ordered pair: (4, 2)xyxxxx-=-=-=-+=+=282 182 82 28 210( ) Ordered pair: (10, 1)ValidateValidatexy-=--==284 2288 8() ?

10 Xy-=-==28102 188 8( ) ?Model 2 Find the x and y- intercepts for the equation x 2y = 8:x-intercept: let y = 0 and solve for xy-intercept: let x = 0 and solve for yxyxxx-=-=-==282 080 88( ) The x -intercept is (8, 0).280 282828224xyyyyy = = = = = The y - intercept is (0, 4).ValidateValidate?xy-=-==2882088 8( )( ) xy-=--==2802488 8( )() ? Sect on . Ordered Pa rs, intercepts , and SlopesModel 3 Calculate the slope for the following equations:EquationFind two Ordered pair solutionsCalculate the slopeValidate2x y = 2 Let x = 12 122200( )-=-=-==yyyyOrdered pair 1(1, 0)Let y = 2222242xxx-=== Ordered pair 2(2, 2)myyxxm=--=--==()()()()21212 02 1slope212 Let x = 42 42822 866( )-=-=-=--=-=yyyyyOrdered pair (4, 6)myyxxm=--=--==()()()()21216 24 22slope42 y = 3x + 2 Let x = 3yxyy=+=+=+=323 329 2 11( ) Ordered pair 1(3, 11)Let y = 44 324 322 33223=+=+===xxxxxOrdered pair 22, 43 2121()()(4 11)233( 7)2933( 7)73731773377317yymxxmmmmm = = = = = = = =21slope37 Let x = 1yyy=+=+=3 123 25( ) Ordered pair (1, 5)myyxxm=--=--=--==()()()()()()