Question

I don't understand how to solve this equation using "Solution by Substitution". I searched up the answer but I don't understand how to get the answer.

Answer 1

(x, y) = (-2, -7)

Explanation:

You want to use "substitution" to solve the system of equations ...

• y = 6x +5
• 2x +y = -11

Substitution

In math, as elsewhere in life, one thing can be substituted for another when the two are completely equivalent. In math, an equivalent is easily found: the equal sign tells you what it is.

The first equation ...

y = 6x +5

tells you that the expressions "y" and "6x+5" are completely equivalent. This is handy. When we want to eliminate y from an equation, we can use "6x+5" instead of "y".

Making this substitution in the second equation gives ...

2x +(6x+5) = -11 . . . . . . the variable y is gone. In its place is (6x+5).

8x = -16 . . . . . . . . . . subtract 5 and simplify

x = -2 . . . . . . . . . . divide by 8

Now, we can find y by using the expression that tell us how it relates to x:

y = 6x +5 = 6(-2) +5 = -12 +5 . . . . . substitute -2 for x

y = -7

The solution to the equation is (x, y) = (-2, -7).

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Additional comment

"Substitution" as a method for solving equations works best if you already have an expression for one variable in terms of the other(s).

Sometimes, you have to solve for the variable you want an expression for. If we were to use the first equation to write an expression for x, we would have to solve for it:

y = 6x +5

y -5 = 6x

(y -5)/6 = x

Using this to substitute for x will give the same answer, possibly requiring a little more work to deal with the fractions. That is why we prefer to look for variables that have coefficients of 1 or -1, or that are already alone on one side of the equal sign.

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