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Solving Systems of Equations: Elimination with Multiplication

Learn how to solve systems of equations using elimination with multiplication. Find the LCM of variables, determine which variable to eliminate, and solve the equations. Check your solutions for accuracy.

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Solving Systems of Equations: Elimination with Multiplication

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  1. Warm Up • What is the LCM of 3x and –4x ? • What is the LCM of 5y and 2y ?

  2. Warm Up • What is the LCM of 3x and –4x ? -12x • What is the LCM of 5y and 2y ? 10y

  3. Objective The student will be able to: solve systems of equations using elimination with multiplication. Designed by Skip Tyler, Varina High School Modified by Lisa Hoffmann Troy Buchanan High School

  4. Solving Systems of Equations • So far, we have solved systems using graphing, substitution, and elimination. These notes go one step further and show how to use ELIMINATION with multiplication. • What happens when the coefficients are not the same? • We multiply the equations to make them the same! You’ll see…

  5. Solving a system of equations by elimination using multiplication. Standard Form: Ax + By = C Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Look for variables that have the same coefficient. (MAKE THEM) Step 3: Multiply the equations and solve. Solve for the variable. Step 4: Plug back in to find the other variable. Substitute the value of the variable into the equation. Step 5: Check your solution. Substitute your ordered pair into BOTH equations.

  6. 1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 Step 1: Put the equations in Standard Form. They already are! None of the coefficients are the same! Find the least common multiple of each variable. LCM = 6x, LCM = 2y Which is easier to obtain? 2y(you only have to multiplythe bottom equation by 2) Step 2: Determine which variable to eliminate.

  7. 1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 Multiply the bottom equation by 2 2x + 2y = 6 (2)(3x – y = 5) 8x = 16 x = 2 2x + 2y = 6 (+) 6x – 2y = 10 Step 3: Multiply the equations and solve. 2(2) + 2y = 6 4 + 2y = 6 2y = 2 y = 1 Step 4: Plug back in to find the other variable.

  8. 1) Solve the system using elimination. 2x + 2y = 6 3x – y = 5 (2, 1) 2(2) + 2(1) = 6 3(2) - (1) = 5 Step 5: Check your solution. Solving with multiplication adds one more step to the elimination process.

  9. 2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 Step 1: Put the equations in Standard Form. They already are! Find the least common multiple of each variable. LCM = 4x, LCM = 12y Which is easier to obtain? 4x(you only have to multiplythe top equation by -4 to make them opposites) Step 2: Determine which variable to eliminate.

  10. 2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 Multiply the top equation by -4 (-4)(x + 4y = 7) 4x – 3y = 9) y = 1 -4x – 16y = -28 (+) 4x – 3y = 9 Step 3: Multiply the equations and solve. -19y = -19 x + 4(1) = 7 x + 4 = 7 x = 3 Step 4: Plug back in to find the other variable.

  11. 2) Solve the system using elimination. x + 4y = 7 4x – 3y = 9 (3, 1) (3) + 4(1) = 7 4(3) - 3(1) = 9 Step 5: Check your solution.

  12. What is the first step when solving with elimination? • Add or subtract the equations. • Multiply the equations. • Plug numbers into the equation. • Solve for a variable. • Check your answer. • Determine which variable to eliminate. • Put the equations in standard form.

  13. Which variable is easier to eliminate? • x • y • 6 • 4 3x + y = 4 4x + 4y = 6

  14. 3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 Step 1: Put the equations in Standard Form. They already are! Find the least common multiple of each variable. LCM = 12x, LCM = 12y Which is easier to obtain? Either! I’ll pick y because the signs are already opposite. Step 2: Determine which variable to eliminate.

  15. 3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 Multiply both equations (3)(3x + 4y = -1) (4)(4x – 3y = 7) x = 1 9x + 12y = -3 (+) 16x – 12y = 28 Step 3: Multiply the equations and solve. 25x = 25 3(1) + 4y = -1 3 + 4y = -1 4y = -4 y = -1 Step 4: Plug back in to find the other variable.

  16. 3) Solve the system using elimination. 3x + 4y = -1 4x – 3y = 7 (1, -1) 3(1) + 4(-1) = -1 4(1) - 3(-1) = 7 Step 5: Check your solution.

  17. What is the best number to multiply the top equation by to eliminate the x’s? • -4 • -2 • 2 • 4 3x + y = 4 6x + 4y = 6

  18. Solve using elimination. • (2, 1) • (1, -2) • (5, 3) • (-1, -1) 2x – 3y = 1 x + 2y = -3

  19. Airplane Speed It took 3 hours for a plane, flying against the wind, to travel 900 miles from Alabama to Minnesota. The “ground speed” of the plane is 300 miles per hour. On the return trip, the flight took only 2 hours with a ground speed of 450 miles per hour. During both flights the speed and the direction of the wind were the same. The plane’s speed decreases or increases because of the wind as shown below: Speed in Still Air – Wind Speed = Ground Speed Against Wind Speed in Still Air + Wind Speed = Ground Speed With Wind

  20. Airplane Speed Speed in Still Air – Wind Speed = Ground Speed Against Windd Speed in Still Air + Wind Speed = Ground Speed With Wind S = Speed in Still Air W = Wind Speed Distance = 900 Ground Speed There = 300 mph Ground Speed Back = 450 mph S – W = 300 S + W = 450 S – W = 300 S + W = 450 375 – W = 300 W = 75 Speed in Still Air = 375 mph Wind Speed = 75 mph 2S = 750 S = 375

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