In order to understand how to solve a two-step equation, it's important that you first understand how to solve a one-step equation. When solving an equation, the goal is to get the variable by itself. In a one-step equation, you use the inverse operation to do this. Addition and subtraction are inverses - they can cancel out what the other one has done. Multiplication and division are also inverses - they can cancel each other out as well. Let's look at one quick example:
4 is being subtracted from the variable x. How do we get x by itself? We need to use the inverse of subtraction and use addition. To cancel out the minus 4, we need to add 4 to both sides. Remember to always do the same thing to BOTH sides of the equation when solving so that you keep things balanced.
It's pretty easy to check your answer. Plug 17 in for x in the original equation. Does 17 - 4 equal 13? Yes. Make sure to check out the lesson on solving one-step equations if you need extra review.
Solving Two-Step Equations
Let's say I have a mystery number. If I multiply it by 2 and then add 5, I get 11. Can you figure out what the number is? Let's write this situation as an equation using the variable x. If I multiply it by 2, I can represent that as 2x. To show that I add 5 next, I can write 2x + 5. This comes out to 11 so my equation is:
How do you solve for x? You need to work your way backwards to get x by itself. The last step was to add 5, so we need to undo that first. How do we undo adding 5? We need to use the inverse operation and subtract 5 from both sides. When solving a two-step equation, you will always need to undo the addition and subtraction first.
Why do you undo addition and subtraction first? Let's say you knew what x was. If you plugged it in, you would have to follow the order of operations to simplify. A lot of teachers use PEMDAS, or Please Excuse My Dear Aunt Sally to remember the order. If you follow the order of operations, you need to multiply and divide before you add or subtract. When you're simplifying, the addition and subtraction are always done last. When you're solving, you're working your way backwards to solve for x. This means you'll need to undo the addition and subtraction first.
Now that we've canceled out the addition, we're left with the equation 2x = 6. The 2 in front of the x means that x is being multiplied by 2. To get x by itself, we need to undo this multiplication. How do you undo multiplying? You need to divide. We can can cancel out the 2 by dividing both sides by 2. Remember to always do the same thing to BOTH sides to make sure the equation stays balanced.
How do you check your answer? Plug 3 in for x in the original equation and make sure both sides of the equation come out to the same number.
Let's try another example. Let's say I have another mystery number. If I divide it by 4 and then I subtract 3 from it, I get 2. What is my number? First, let's write this as an equation. If we call the mystery number x, we could use x/4 to represent dividing the number by 4 and then put the minus 3 after it to show that we subtracted 3 next. This gives us the equation:
When you're solving an equation, the goal is to get the variable by itself. We need to work our way backwards to figure out what x was. Remember, we divided first and then subtracted. If we go backwards to find x, the first thing we need to do is to undo the subtraction. How do you undo subtracting 3? We need to use the inverse operation to cancel it and add 3 to both sides. Remember to always do the same thing on both sides of the equation so everything stays balanced.
Why do you undo the subtraction first? If you knew what x was and you were using PEMDAS to simplify the expression, you would do the subtraction last. When you're solving, you're working your way backwards to get x by itself. Since addition and subtraction are done last when simplifying, that means they'll be the first things you undo when you're solving an equation.
Now that we've canceled out the minus 3, we're left with x/4 = 5. This means that x is being divided by 4. How do we undo division? We need to use the inverse operation and multiply. If we multiply both sides by 4, it will cancel the division by 4. Remember to always do the same thing on both sides of the equation. A common mistake is for students to multiply one side and forget to multiply the other side.
How do you check your answer to see if it's right? You plug your answer back in the original equation and see if it works. We need to plug 20 in for x in the original equation and make sure it comes out to 2. If we divide 20 by 4 and then subtract 3, we see that it does come out to 2. This means we have the correct answer.