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LESSON 3: USING A SUBSTITUTION STRATEGY TO SOLVE A SYSTEM OF LINEAR EQUATIONS Learning Outcome: Learn to use the substitution of one variable to solve a linear system. Work with a partner. Solve each linear system without graphing: 3x + 5y = 6 x = -4 2x + y = 5 y = -x + 3 What strategies did you use? How can you check that each solution is correct? In the previous lessons, we solved linear systems by graphing. This strategy is time-consuming even when you use graphing technology, and you can only approximate the solution. We use algebra to determine an exact solution. One algebraic strategy is called solving by substitution. By using substitution, we transform a system of two linear equations into a single equation in one variable, then we use what we know about solving linear equations to determine the value of that variable. The skill of substituting algebraic expressions is used regularly in math and science. The substitution method can provide a quick solution to a linear system. Solve the following system: 4x + 5y = 26 3x = y – 9 Ex. Solve this linear system: 5x – 3y = 18 4x – 6y = 18 Ex. Admission to the Abbotsford International Airshow cost $80 for a car with two adults and three children. Admission for a car with two adults cost $50. Use algebra to determine the cost for one child and the cost for one adult. There was no charge for the vehicle or parking. These two linear systems have the same graph and the same solution x = 1 and y = 2. System A 𝑥 2 1 3 +𝑦 = System B 5 Multiply each term by 2 2 1 1 Multiply each term by 3 3 3 𝑥− 𝑦=− 𝑥 + 2𝑦 = 5 𝑥 − 𝑦 = −1 In system A, we can multiply the first equation by 2, and the second by 3 to write equivalent equations with integer coefficients. The result is the equations in system B. The two linear systems have the same solution (1, 2) because the corresponding equations are equivalent. Multiplying or dividing the equations in a linear system by a non-zero number does not change the graphs. So, their point of intersection, and hence, the solution of the linear system is unchanged. A system of equivalent equations is called an equivalent linear system and has the same solution as the original system. When an equation in a linear system has coefficients or a constant term that are fractions, we can multiply by a common denominator to write an equivalent equation with integer coefficients. Ex. Solve this linear system by substitution. 1 2 4 𝑥 − 𝑦 = −2 5 1 3 4 8 𝑦= 𝑥− Assignment: pg. 424-427 #1-22, 27