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how to do x-y=2 and4x-3y=11 doing substitution

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3 Answers

Hi Ciera,

This can be solved in a couple of steps.

Do you know how to "isolate" a variable? You know, get a variable on one side by itself? This is step one. Choose one of the equations, doesn't matter which, and solve for one of the variables (doesn't matter which).

For Example, if you chose this equation: x - y = 2

You could solve for x by adding y to both sides. This would eliminate the y variable on the left side and add it to the right side... giving you this: x = 2 + y

Now you have isolated the variable, x.

Take your new equation, x = 2 + y and substitute this equation into the second equation.

Notice that the second equation has both x and y variables also: 4x - 3y = 11

You can replace (substitute) your first equation into the position of the 'x' in the second equation. Why? So that you will only have one variable in the second equation instead of two. Then you'll be able to solve the second equation and find out the value of y. It looks like this:

4x - 3y = 11

4(2 + y) - 3y = 11

Notice that every other part of the second equation remains the same. The only thing we changed was that we replaced the x variable with part of the first equation. Since x = 2 + y, we can substitute the x for 2 + y in the second equation.

Now we solve for y by following these steps: distribute, add like terms, and isolate the y variable. It looks like this:

Given: 4(2 + y) - 3y = 11

Distribute: 4(2) + 4y - 3y = 11

Multiply: 8 + 4y - 3y = 11

Combine like terms (4y - 3y = y): 8 + y = 11

Isolate the variable by subtracting the 8 from both sides: y = 11 - 8

Subtract: y = 3

Now that you know the value of y, you can substitute that value into the first equation to find the value of x.

x = 2 + y

x = 2 + 3

x = 5

The solution to the system of equations is x = 5 and y = 3

Written as a coordinate: (5, 3)

Substitution? Easy as pie! Think of it this way, when you substitute something, you're replacing it with something else, right? That's what substitution in math is about too. You replace a letter you don't know with something you do know, so you can make the problem easy to solve.

Step 1: Write out both equations. 

Step 2: Make one of them equal to y, by moving things numbers around (use the opposites to move them from one side of the equation to the other.)

Step 3: Go to the second equation and plug in the part of the first equation that is after the equal sign.

Step 4: Solve the second equation by adding or subtracting like terms. 

Step 5: Plug in that answer into the first equation and solve for the second unknown number (variable).

That's all there is to it...well, besides checking to make sure they both equal the numbers on the other side of the equations! 

Let's try your example:

x-y=2 and4x-3y=11

x-y=2

-x from both sides of the equation.

You are now left with -y=2-x

Divide by -1 to get y alone. 

Do the same thing to the other side. Divide by -1

When you divide by a negative 1, all the signs change to the opposite signs. 

So, y = -2+x

plug that into the second equation. Wherever you see y in the second equation, replace it with the -2+x.

Then, solve the second equation.

 4x-3y=11

4x-3(-2+x)=11

Use distribution to solve for the parenthesis part and you will have 4x+6-3x=11

Add like terms: 4x-3x=1x or just x

You will now be left with: x+6=11

Subtract 6 from both sides. 

x+6-6=11-6

x=5

Plug that answer back into the first original equation and solve for y. 

x-y=2

5-y=2

subtract 5 from both sides: 5-5-y=2-5

-y=-3

divide by -1 and you will have y = 3

Now, you just check to make sure it all works out, by plugging both answers back into both equations and seeing if they equal the totals already in the original equations. 

For instance: x-y=2

5-3=2

2=2  check

Does it check? Yes, it does! 

Do the same with the other one and presto...you're a math whiz! ;)  

What is meant by substitution is that if A=B, then in any equation with A, every occurrence of A can be replaced by B. In this casewe can start by isolating x in the first equation:

x - y = 2

x = y + 2

Now in the second equation, we can replace every x with a y+2 (parentheses are important here) and solve for y:

4(y+2) - 3y = 11

4y + 8 - 3y = 11 (Used distributive property)

y + 8 = 11 (Collected like terms)

y = 3

Then solve for x:

x = y+2 = 3+2 = 5

So the solution to the system is x=5, y=3.