How do you solve each system by substitution or addition y equals 2x-9 and x 3y equals 8?

Answer 1

To solve a system of equations by substitution:

y = 2x-9

x + 3y = 8

We need to solve ONE equation for a variable. You choose, X or Y.

When choosing which equation, pick one without numbers (coefficients) in front of X or Y.

(Hint: Take a look at the top equation... it's already solved for Y!)

Since y = 2x - 9 is already solved for y, let's substitute y's value, 2x - 9 in for wherever we see the letter y in the 2nd equation. Remember to use parenthesis!

x + 3y = 8

x + 3(2x -9) = 8

Now use your algebra to simplify.

x + 6x - 27 = 8 (Distributive property)

7x - 27 = 8 (combine like terms, x and 6x)

7x = 8 + 27

7x = 35

x = 5.

We now know that x = 5. So let's use substitution again!

Plug 5 in for x, wherever you see x in your original equation.

We need to find y now!

y = 2x - 9

y = 2(5) - 9

y = 10 - 9

y = 1.

So x = 5, and y = 1 in this system of equations.

Plug 5 in for x and 1 in for y into both equations to check.

y = 2x-9

1 = 2(5) - 9

1 = 10 - 9

1 = 1 (check)

x + 3y = 8

5 + 3(1) = 8

5 + 3 = 8

8 = 8 (check)

---- To solve a system of equations by elimination:

(Sometimes elimination is also called solving by addition or subtraction.)

Our goal with this method is to eliminate, or get rid of, one variable.

Just like before, you pick the variable. I'll choose X.

Solve each linear equation so that your x's, y's and constant terms are all in the same place. An easy way to do this is to put each equation into Standard Form.

Standard form for linear equations will look like: y = mx +b.

Where, m is your slope and b is the y-intercept.

y = 2x - 9 is already in standard form, so let's work on the other equation.

x + 3y = 8

3y = -x + 8

y = -1/3 x + 8/3

Already we have fractions, and I can tell that elimination is not the *best* method to solve this system, but it's still doable.

Line both equations up, one on top of the other like an addition problem. Make sure the y's and x's line up.

y = 2x - 9

y = -1/3x + 8/3

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Now, we need to add OR subtract.

You ADD the equations together IF AND ONLY IF :

1) The variable being eliminated has the same coefficient for both equations.

2) The variable being eliminated has the OPPOSITE sign in both equations.

You SUBTRACT the equations together IF AND ONLY IF :

1) The variable being eliminated has the same coefficient for both equations.

2) The variable being eliminated has the SAME sign in both equations.

In our equations, the y's have no numbers in front (no coefficient), so #1 is OK.

But the signs are the same, this tells us we need to subtract the bottom equation.

To do this easily, just change the sign of everything in the 2nd equation. Then subtract like terms, straight down.

y = 2x - 9

-y=1/3x - 8/3

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0 = 7/3x - 35/3

Now that y has been eliminated, which was our goal, solve the equation for x.

35/3 = 7/3x

(3/7)(35/3) = x

35/7 = x

x = 5.

Hey look, x = 5. That's the same as our first method! We must be doing it right :D

Now all we need to do is solve for y. You don't need to use elimination again. Just use substitution!

Plug x = 5 into either equation and solve for y.

y = 2x - 9

y = 2(5) - 9

y = 10 - 9

y = 1.

X = 5 and Y = 1 again, good job!

As you can see, either method will get you the right answers. It's up to you to pick which method is best. Which one you prefer to use. (I like using substitution, myself)