Solve Systems of Equations with the Substitution Method

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The substitution method is my favorite way to solve systems of equations. It will always work—it is actually quite mechanical. The answer will always appear, as long as you follow the recipe. It’s an advantage to set up the equations in separate columns when you use the substitution method.

Rule

Recipe for the Substitution Method

1.
Write the equations next to each other in two columns.
2.
Choose one of the equations and solve for one of the variables.
3.
Substitute the expression for that variable into the equation you haven’t used yet. Solve that equation.
4.
Put the answer back into the first equation and solve for the remaining variable.
5.
Write the answer in coordinate form: ANSWER: (x,y) = (a,b).

A system of equations concerning the amount of goals Ronaldo and Messi scored in a season.

Example 1

Solve this system of equations with the substitution method:

y + 2x = 1 (1) y x = 2 (2)
1.
Write down the equations:

(1)y + 2x = 1 (2)y x = 2

(1)y + 2x = 1(2)y x = 2

2.
Choose the equation you think looks easier and solve for one of the variables: (1)y + 2x = 1 y = 1 2x
3.
Insert the expression you just found into the equation you haven’t used yet: y x = 2 (1 2x) x = 2 1 2x x = 2 3x = 2 1 3x 3 = 3 3 x = 1
4.
Put this value back into the first expression you found: y = 1 2x y = 1 2 (1) y = 1 2 y = 1
5.
Write the answer in coordinate form:
(x,y) = (1,1)

Example 2

Mom, Aunt Beth, and the three kids are looking forward to watching the world champion Magnus Carlsen play in a chess tournament in London. They have to pay £550 in total for their tickets.

David Beckham also wants to watch the same chess tournament. He and his wife Victoria will be joined by one of David’s buddies, and his oldest son. They pay a total of £370 for their tickets. Find the price of an adult ticket and a child ticket.

The unknowns in this system is the price of an adult ticket and the price of a child ticket. Let’s call the adult ticket x and the child ticket y.

The equation describing the situation of Mom, Aunt Beth, and the three kids looks like this:

2x + 3y = 550
(3)

The equation for David Beckham and his companions looks like this:

3x + y = 370
(4)

You choose to start with Equation (4), because that equation has a variable that stands on its own, y.

3x + y = 370, y = 370 3x

Apply this expression to Equation (3) and solve for x:

2x + 3y = 550 2x + 3(370 3x) = 550 2x + 1110 9x = 550 7x = 560 x = 80.

2x + 3y = 550 2x + 3(370 3x) = 550 2x + 1110 9x = 550 7x = 560| ÷ (7) x = 80.

You can now put this value for x back into (4) to find y:

y = 370 3 80 = 130

Surprisingly, the adult tickets are actually cheaper than the child tickets. Adult tickets cost £80 and child tickets cost £130.

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