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QUESTION

# What is the final step of completing a solve by substitution problem?

I'm not sure where exactly you mean "final" in the solving process. So, I've prepared a couple of problems that I will through slowly and carefully, showing all the steps to the final answer.

Example 1: Solve the following system of equations-2x + y = 5, 3x + 2y = 9

Since we want to solve with substitution, we must solve for one variable in one of the equations. I think it would be easiest to solve for y in the first equation.

y = 5 - 2x

We can now substitute into the other equation:

3x + 2(5 - 2x) = 9

3x + 10 - 4x = 9

-x = -1

x = 1

We must now find the value of y. This is found by inserting x = 1 into one of the equations and solving for y.

y = 5 - 2x

y = 5 - 2(1)

y = 3

Hence, our solution set is {1, 3}.

Example 2: Find all real values of x and y that satisfy the following system of equations: 3y = -2x^2 + 2, 2x^2 - 3y^2 = -4

Once again, as with the last example, we need to solve for one of the variables in one of the equations. It looks easiest to isolate the y in the first equation, however solving this equation won't be as neat as solving the previous one.

y = -2/3x^2 + 2/3

We can now substitute into equation 2.

2x^2 - 3(-2/3x^2 + 2/3)^2 = -4

2x^2 - 3(4/9x^4 - 8/9x^2 + 4/9) = -4

2x^2 - 4/3x^4 + 8/3x^2 -4/3 = -4

-4/3x^4 +14/3x^2 + 8/3 = 0

Solve using a graphing calculator. If it's a standard one, like a TI84, use y_1 = -4/3x^4 + 14/3x^2 + 8/3 = 0 and y_2 = 0, and press "calc" followed by "intersect".

This will give you real roots of 2 and -2. All that is left to do is solve for y.

y = -2/3(2)^2 + 2/3" AND "-2/3(-2)^2 + 2/3

y = -8/3 + 2/3" AND "-8/3 + 2/3

y = -2" AND " -2

Hence, our solution set is {2, -2} and { -2, -2}.

Use the following practice exercises to develop your comfort with the skills dealt with in this answer.

Practice exercises:

1. Find the real values of x and y that satisfy the following systems of equations.

a) 2x - 3y = 4, x + 2y = 9

b) 3x + y = -2, x^2 = y

c) 2x^2 - 3y^2 = -10, x^2 - 2x + 3y = 5

Hopefully this helps, and good luck!