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3-2 3-2 Using UsingAlgebraic AlgebraicMethods Methods totoSolve SolveLinear LinearSystems Systems Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 22 3-2 Using Algebraic Methods to Solve Linear Systems Lesson Quiz: Part I Solve the system using your graphing calculator. 1. x+y=1 3x –2y = 8 (2, –1) 2. Lynn is piloting a plane at an altitude of 10,000 ft. She begins to descend at a rate of 200ft per minute. Miguel is flying a different plane at an altitude of 5000 ft. At the same time that Lynn begins to descend, Miguel begins to Y = 10000-200x climb at a rate of 50 ft. per minute. A. Write and graph a system of equations that could be Y = 5000 + 50x used to model the situation. B. In how many minutes will the planes be at the same altitude? 20 min C. What will the altitude be? 6000 ft Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Objectives Solve systems of equations by substitution. Solve systems of equations by elimination. Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Example 1A: Solving Linear Systems by Substitution Use substitution to solve the system of equations. y= x–1 x+y=7 Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Check It Out! Example 1a Use substitution to solve the system of equations. y = 2x – 1 3x + 2y = 26 Step 1 Solve one equation for one variable. The first equation is already solved for y: y = 2x – 1. Step 2 Substitute the expression into the other equation. 3x + 2y = 26 Substitute (2x –1) for y in 3x + 2(2x–1) = 26 the other equation. 3x + 4x – 2 = 26 Combine like terms. 7x = 28 x=4 Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Check It Out! Example 1a Continued Step 3 Substitute the x-value into one of the original equations to solve for y. y = 2x – 1 y = 2(4) – 1 Substitute x = 4. y=7 The solution is the ordered pair (4, 7). Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems You can also solve systems of equations with the elimination method. With elimination, you get rid of one of the variables by adding or subtracting equations. You may have to multiply one or both equations by a number to create variable terms that can be eliminated. Reading Math The elimination method is sometimes called the addition method or linear combination. Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Example 2A: Solving Linear Systems by Elimination Use elimination to solve the system of equations. 3x + 4y = 3 4x – 2y = –18 Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Check It Out! Example 2a Use elimination to solve the system of equations. 2x + 6y = –8 5x –3y = 88 Step 1 Find the value of one variable. 2x + 6y = –8 10x – 6y = 176 12x = 168 x = 14 Holt Algebra 2 The y-terms have opposite coefficients. Add the equations to eliminate y. First part of the solution 3-2 Using Algebraic Methods to Solve Linear Systems Check It Out! Example 2a Continued Step 2 Substitute the x-value into one of the original equations to solve for y. 2(14) + 6y = –8 28 + 6y = –8 6y = –36 y = –6 Second part of the solution The solution to the system is (14 , –6). Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems In Lesson 3–1, you learned that systems may have infinitely many or no solutions. When you try to solve these systems algebraically, the result will be an identity or a contradiction. Remember! An identity, such as 0 = 0, is always true and indicates infinitely many solutions. A contradiction, such as 1 = 3, is never true and indicates no solution. Holt Algebra 2 Using Algebraic Methods 3-2 to Solve Linear Systems Example 3: Solving Systems with Infinitely Many or No Solutions Classify the system and determine the number of solutions. 3x + y = 1 2y + 6x = –18 Because isolating y is straightforward, use substitution. 3x + y = 1 y = 1 –3x Solve the first equation for y. 2(1 – 3x) + 6x = –18 Substitute (1–3x) for y in the second equation. 2 – 6x + 6x = –18 Distribute. 2 = –18 x Simplify. Because 2 is never equal to –18, the equation is a contradiction. Therefore, the system is inconsistent and has no solution. Holt Algebra 2 3-2 Using Algebraic Methods to Solve Linear Systems Lesson Quiz Use substitution or elimination to solve each system of equations. 1. 3x + y = 1 y= x+9 (–2, 7) 2. 5x – 4y = 10 3x – 4y = –2 (6, 5) 3. The Miller and Benson families went to a theme park. The Millers bought 6 adult and 15 children tickets for $423. The Bensons bought 5 adult and 9 children tickets for $293. Find the cost of each ticket. adult: $28; children’s: $17 Holt Algebra 2