How to solve systems of equations (two variables, two equations)- Elimination/Addition Method
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1 How to solve systems of equations (two variables, two equations)- Elimination/Addition Method This guide will show you how to solve these equations using the elimination/addition method. The elimination/addition method works by taking both of our equations and multiplying one equation by a number and the second equation by another number such that when these equations are added, we eliminate one of the variables. Example 1 Solve the system of equations x + 2y = 1 3x + 4y = 1 It really does not matter which variable we choose to eliminate. However, sometimes it is easier to eliminate one variable. I will choose x. We multiply the equation x + 2y = 1 by 3 so when added to the equation 3x + 4y = 1, x will be eliminated. Let s go ahead and multiply x + 2y = 1 by 3: 3(x + 2y) = 3 1 will give us 3x 6y = 3 We will now add the equations: 3x 6y = 3 and 3x + 4y = 1: 3x 6y = 3 + 3x + 4y = 1 2y = 4 STEP 3: You should have an equation with only one variable. Solve this equation 2y = 4 2y 2 = 4 2 y = 2
2 STEP 4: Solve for the other variable by plugging in the value you got in STEP 3 to one of the original equations and solve. I will take x + 2y = 1 and replace y with y = 2 and solve x + 2( 2) = 1 x 4 = 1 x = x = 3 STEP 5: Write the answer in an ordered pair: (3, 2) Example 2 Solve the system of equations I will eliminate y 6x + 3y = 21 5x + 7y = 40 I will multiply the first equation by 7 and the second equation by 3. This will give us coefficients of 21 and 21 for y in each equation respectively. Multiplying the first equation by 7: Multiplying the second equation by 3: 7(6x + 3y) = 7 21 will give us: 42x + 21y = 147 3(5x + 7y) = 3 40 will give us: 15x 21y = 120 We will now add the equations: 42x + 21y = 147 and 15x 21y = x + 21y = x 21y = 120
3 27x = 27 STEP 3: You should have an equation with only one variable. Solve this equation 27x = x = 1 STEP 4: Solve for the other variable by plugging in the value you got in STEP 3 to one of the original equations and solve. I will take 6x + 3y = 21 and replace x with x = 1 and solve for y: 6(1) + 3y = y = 21 3y = 15 3y = y = 5 STEP 5: Write the answer in an ordered pair: (1,5) Example 3 Solve the system of equations I will eliminate x. 6x + 5y = 59 4x + 3y = 37 I will multiply my first equation by 2 and the second equation by 3. This will give us coefficients of 12 and 12 for x in each equation respectively. (Note: I could have multiplied the first equation by 4 and the second equation by 6 as well, to get rid of x. However, this would have given us larger numbers to work with.)
4 Multiplying the first equation by 2: 2(6x + 5y) = 2 59 will give us: 12x + 10y = 118 Multiplying the second equation by 3: 3(4x + 3y) = 3 37 will give us: 12x 9y = 111 We will now add the equations: 12x + 10y = 118 and 12x 9y = x + 10y = x 9y = 111 y = 7 STEP 3: You should have an equation with only one variable. Solve this equation We don t need to do anything here since we have solved for y = 7 STEP 4: Solve for the other variable by plugging in the value you got in STEP 3 to one of the original equations and solve. I will take 4x + 3y = 37 and replace y with y = 7 and solve for x: 4x + 3(7) = 37 4x + 21 = 37 4x = x = 16 x = 4 STEP 5: Write the answer in an ordered pair: (4, 7) Example 4 Solve the system of equations x + 2y = 10 2x + 4y = 20
5 I will choose to eliminate y. I will multiply the equation x + 2y = 10 by 2 and add the result to the equation 2x + 4y = 20 Multiplying the equation x + 2y = 10 by 2 will give us: 2x 4y = 20 I will now add 2x 4y = 20 and 2x + 4y = 20: 2x 4y = x + 4y = 20 0 = 0 STOP!!! We don t have any variables left!!! When you have two numbers equal to each other, we have an infinite number of solutions. We are done. Our answer is: infinite number of solutions. Example 5 Solve the system of equations I will choose to eliminate x. x + 2y = 5 2x + 4y = 20 I will multiply the equation x + 2y = 5 by 2 and add the result to the equation 2x + 4y = 20 Multiplying the equation x + 2y = 5 by 2 will give us: 2x 4y = 10 I will now add 2x 4y = 10 and 2x + 4y = 20: 2x 4y = 10
6 + 2x + 4y = 20 0 = 10 STOP!!! We don t have any variables left!!! When you have two numbers that don t equal to each other, we have no solution. We are done. Our answer is: no solution (which we sometimes denote with the symbol
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