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Applications of Systems of Linear Equations Example 1: Steve invested $12,000 for one year in two different accounts, one at 3.5% and the other at 4%. The combined interest on the two accounts was $445. Use a system of equations to determine the amount in each account. Make a drawing to illustrate the situation.

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Part at 3.5% Part at 4% 1) Variable declarations Let x be the amount at 3.5%. Let y be the amount at 4%.

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2) Write the equations. Since this is a system with two variables, we need two equations. Steve invested $12,000 for one year in two different accounts, one at 3.5% and the other at 4%. The combined interest on the two accounts was $445. Use a system of equations to determine the amount in each account. $12,000 is the amount in the combined accounts. (amount at 3.5%) + (amount at 4%) = (total amount)

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Use the interest on the accounts to write the second equation. Steve invested $12,000 for one year in two different accounts, one at 3.5% and the other at 4%. The combined interest on the two accounts was $445. Use a system of equations to determine the amount in each account. 3.5% Account: 4% Account:

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Steve invested $12,000 for one year in two different accounts, one at 3.5% and the other at 4%. The combined interest on the two accounts was $445. Use a system of equations to determine the amount in each account. 3.5% Account: 4% Account: (interest on 3.5%) + (interest on 4%) = (total interest)

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The two equations are … 3) Solve the system: Multiply every term in the second equation by 1000 to eliminate decimals.

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Multiply the top equation by ( - 35).

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Add the equations: Substitute into the first of the original equations:

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4) Write an answer in words, explaining the meaning in light of the application Part at 3.5% Part at 4% Steve invested 7000 at 3.5% and 5000 at 4%.

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Example 2: Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate. Applications of Systems of Linear Equations

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Drt Sam Mary Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate. Let x represent Sam’s rate. 1) Variable declaration: Let y represent Mary’s rate

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Drt Sam Mary Both Sam and Mary were traveling the same amount of time, from 11:00am to 3:00pm, which is 4 hours. Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate.

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Drt Sam Mary Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate. Since distance = rate × time, Sam’s distance is … … and Mary’s distance is…

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Drt Sam Mary Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate. (Sam’s distance) + (Mary’s distance) = 480 miles 2) Write the first equation

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Drt Sam Mary Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate. (Sam’s rate) = (Mary’s rate) - 10 Write the second equation

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3) Solve the system: Substitute the expression in the second equation for x in the first equation.

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Solve the equation:

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4) Write an answer in words, explaining the meaning in light of the application What was asked for in the application Sam leaves a gas station in Denver at 11:00am heading west. At the same time, Mary leaves the same station heading east. Since Sam is driving in the mountains, his average rate is 10 mph slower than Mary’s. At 3:00pm they are 480 miles apart. Determine Sam’s rate.

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y =Mary’s rate Mary’s rate was 65 mph. Sam’s rate was x. Sam’s rate was 55 mph.

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