1.1 Integers and Rational Numbers

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$Lesson 2a – Assessment - Scottsdale Community College - Math Blog$

www.ck12.org Chapter 12. Systems of Linear EquationsWe cant just multiply a single equation by an integer number in order to arrive at matching coefficients. But if welook at the coefficients of x (as they are easier to deal with than the coefficients of y), we see that they both have acommon multiple of 15 (in fact 15 is the lowest common multiple). So we can multiply both equations.Multiply the top equation by 5:15x + 4400y = 4200Multiply the lower equation by -3:−15x − 6180y = −5535Add:15x + 4400y = 4200+ (−15x − 6180y) = −5535− 1780y = −1335Divide by − 1780 : y = 0.75Substitute this back into the top equation:3x + 880(0.75) = 840 since 880 × 0.75 = 660, subtract 660 f rom both sides :3x = 180 divide both sides by 3x = 60I-Haul charges $60 per day plus $0.75 per mile.Comparing Methods for Solving Linear SystemsNow that weve covered the major methods for solving linear equations, lets review them. For simplicity, well lookat them in table form. This should help you decide which method would be best for a given situation.TABLE 12.6:Method: Best used when you. . . Advantages: Comment:Graphing. . . dont need an accurateanswer.Often easier to see numberand quality of intersectionson a graph. Witha graphing calculator, itcan be the fastest methodsince you dont have to doany computation.Can lead to imprecise answerswith non-integer solutions.255
12.3. Solving Linear Systems by Elimination www.ck12.orgTABLE 12.6: (continued)Method: Best used when you. . . Advantages: Comment:Substitution. . . have an explicit equationfor one variable (e.g.Works on all systems. Reducesthe system to oneYou are not often givenexplicit functions in systemsy = 14x + 2)variable, making it easierproblems, so youto solve.may have to do extra workto get one of the equationsElimination by Additionor SubtractionElimination by Multiplicationand then Additionand Subtraction. . . have matching coefficientsfor one variable inboth equations.. . . do not have any variablesdefined explicitly orany matching coefficients.Easy to combine equationsto eliminate onevariable. Quick to solve.Works on all systems.Makes it possible to combineequations to eliminateone variable.into that form.It is not very likely thata given system will havematching coefficients.Often more algebraic manipulationis needed toprepare the equations.The table above is only a guide. You might prefer to use the graphical method for every system in order to betterunderstand what is happening, or you might prefer to use the multiplication method even when a substitution wouldwork just as well.Example 7Two angles are complementary when the sum of their angles is 90 ◦ . Angles A and B are complementary angles, andtwice the measure of angle A is 9 ◦ more than three times the measure of angle B. Find the measure of each angle.SolutionFirst we write out our 2 equations. We will use x to be the measure of angle A and y to be the measure of angle B.We get the following system:x + y = 902x = 3y + 9First, well solve this system with the graphical method. For this, we need to convert the two equations to y = mx + bform:x + y = 90 ⇒ y = −x + 902x = 3y + 9 ⇒ y = 2 3 x − 3The first line has a slope of -1 and a y−intercept of 90, and the second line has a slope of 2 3and a y−intercept of -3.The graph looks like this:256
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1.1 Integers and Rational Numbers

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