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## 8th grade (Illustrative Mathematics)

### Unit 4: Lesson 12

Lesson 14: Solving more systems- Systems of equations with substitution: 2y=x+7 & x=y-4
- Systems of equations with substitution
- Systems of equations with substitution: y=4x-17.5 & y+2x=6.5
- Systems of equations with substitution
- Systems of equations with substitution: y=-5x+8 & 10x+2y=-2
- Substitution method review (systems of equations)

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# Systems of equations with substitution

Walk through examples of solving systems of equations with substitution.

Let's work to solve this system of equations:

The tricky thing is that there are two variables, x and y. If only we could get rid of one of the variables...

Here's an idea! Equation 1 tells us that start color #e07d10, 2, x, end color #e07d10 and start color #e07d10, y, end color #e07d10 are equal. So let's plug in start color #e07d10, 2, x, end color #e07d10 for start color #e07d10, y, end color #e07d10 in Equation 2 to get rid of the y variable in that equation:

Brilliant! Now we have an equation with just the x variable that we know how to solve:

Nice! So we know that x equals 8. But remember that we are looking for an ordered pair. We need a y value as well. Let's use the first equation to find y when x equals 8:

Sweet! So the solution to the system of equations is left parenthesis, start color #11accd, 8, end color #11accd, comma, start color #1fab54, 16, end color #1fab54, right parenthesis. It's always a good idea to check the solution back in the original equations just to be sure.

Let's check the first equation:

Let's check the second equation:

Great! left parenthesis, start color #11accd, 8, end color #11accd, comma, start color #1fab54, 16, end color #1fab54, right parenthesis is indeed a solution. We must not have made any mistakes.

Your turn to solve a system of equations using substitution.

## Solving for a variable first, then using substitution

Sometimes using substitution is a little bit trickier. Here's another system of equations:

Notice that neither of these equations are already solved for x or y. As a result, the first step is to solve for x or y first. Here's how it goes:

**Step 1: Solve one of the equations for one of the variables.**

Let's solve the first equation for y:

**Step 2: Substitute that equation into the other equation, and solve for x.**

**Step 3: Substitute x, equals, 4 into one of the original equations, and solve for y.**

So our solution is left parenthesis, start color #11accd, 4, end color #11accd, comma, start color #1fab54, 3, end color #1fab54, right parenthesis.

## Let's practice!

## Want to join the conversation?

- Props to all the homies in the struggle (the comment section) actually asking and answering questions that are relevant to the topic full heartedly. Just a little thank you.(27 votes)
- welp this helped me a little bit(16 votes)
- x=y+2,55

2x=5y

Substitute please(10 votes)- x = y + 2.55

2x = 5y

By substituting the first equation to the second equation we get:

2(y+2.55) = 5y

Now distribute:

2y + 2.55 = 5y

Then we combine like terms:

2.55 = 5y - 2y

which is 2.55 = 3y.

We then finally divide 3 and 2.55 to get y.

2.55/3 = y

y = 0.85. So y equals 0.85. Now we can find x easily.

x = 0.85 + 2.55

x = 3.40(2 votes)

- In the first question when the ask you 4x+y=28 and the second equation is y=3x what do i do, do i plug in 3x into the first equation please help?(4 votes)
- Yes so it would become 4x + (3x) + 28 then, 7x = 28 and finally you would get x = 4(7 votes)

- i am still stuck is it another way...?(7 votes)
- When do we need to use this in real life?(3 votes)
- Good question. It will likely only matter if you go into a hard science field with lots of math.(9 votes)

- -5=5x+5y when you divide it by 5 I don't understand how the equation then becomes x=-1-y when you are dividing by a positive 5 :((5 votes)
- Two things are happening - first, divide by positive 5, then manipulate the equation to get x by itself.

So take the original equation and divide each term on both sides by positive 5. As long as we divide each term by the same non-zero number, the two sides will remain equal.`-5 = 5x + 5y`

-1 = x + y

Now to get x by itself, we can subtract y from both sides.`-1 = x + y`

-1 - y = x + y - y

-1 - y = x

Hope that helps.(2 votes)

- this is so confusing i dont know how to do anything(5 votes)
- What if you solve a system of equations and it ends up as y=y what does that mean(3 votes)
- If you get an equation that is always true, then there are infinitely many solutions (that is, the two original equations are equivalent).(4 votes)

- the last equation definitely got me lost,so in the first part of solving the equation, if you switch

-1=-x+y then "x=-1+y", ain't it supposed to be "x=1+y"?(4 votes)