Question

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

Answer 1

I'm not sure where exactly you mean "final" in the solving process. So, I've prepared a couple of problems that I will work 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!