Solving Algebraic Equations A Quick Guide

Curriculum Expectations Overall Expectations By the end of Grade 7, students will: solve simple algebraic equations using a variety of strategies, including inspection and guess and check. By the end of Grade 8, students will: solve and verify algebraic equations, using a variety of strategies, including inspection, guess and check, and using a “balance” model. Curriculum Expectations

Solving Algebraic Equations by Inspection Solving algebraic equations by inspection simply means that we can tell what the answer is just by looking at the equation. For example, x + 2 = 5 We know that 3 + 2 =5, therefore x =3.

Solving Algebraic Equations Using Guess and Check When we solve an equation using guess and check, we estimate what we think the answer might be, then substitute it into the equation to see how close we are. For example, 2x – 4 = 18 We could estimate the x = 10. If we substitute this into the equation ...

2x – 4 = 18 2 (10) – 4 = 18 20 – 4 = 18 16 ≠ 18 Since 16 does not equal 18, we know that the answer is wrong but since 16 is close to 18, we know that the answer is close to 10.

We can try another “guess. ” Let’s try 12 We can try another “guess.” Let’s try 12. 2x – 4 = 18 2 (12) – 4 = 18 24 – 4 = 18 20 ≠ 18 Again, we know the answer is wrong, however, 18 is between 16 and 20, therefore, x must be between 10 and 12.

The answer is probably 11. Let’s “check” to make sure The answer is probably 11. Let’s “check” to make sure. 2x -4 = 18 2(11) – 4 = 18 22 – 4 = 18 18 = 18 Since the left side is equal to the right side, x = 11.

Solving Equations using the Balance Model When we solve an equation using the balance model, we want the left side of the equation to equal the right side, just like a balance.

When we are solving an equation, we want to “isolate the variable,” which means we want to get the letter on one side of the equation (usually the left) all by itself. For example, 3x + 6 = 21 We eventually want our answer to look like x = ? where only the variable (in this case, x) remains on the left side of the equal sign.

We must ALWAYS remember that whatever we do to one side of the equation, we must do to the other side as well, so that our equation remains balanced.

Solving Equations using Addition and Subtraction When we have an equation that has addition or subtraction in it, we use the opposite operation to help us solve the equation. For example, x + 2 = 3 Remember we have to isolate the variable (get x all by itself). In order to do this, we would have to “get rid of” the 2.

If we subtract 2 from the left hand side, we would be left with just x, since 2 -2 is 0. x + 2 – 2 = 3 ***But to keep our equation balanced, whatever we do to one side, we must to do to the other. Therefore we must also subtract 2 from the right hand side. x + 2 – 2 = 3 – 2 x = 1

To check if our answer is right, we substitute it into the original equation. X + 2 = 3 1 + 2 = 3 3 = 3 Since the left side equals the right side, we know our answer is correct.

A different way of thinking . . . For addition and subtraction questions, you can also use a different method. We still want to isolate the variable on one side of the equation. We also want to put all the numbers on the other side of the equation. This is the simplest form of collecting like terms, which we will discuss later.

X + 2 = 3 Using this method, we are going to bring the 2 over to the other side of the equation. Whenever you bring a number or a variable across the equal sign, you must change it’s sign, meaning if it positive, you make it negative and vice versa. Since 2 is positive, we will make it negative when we bring it across to the right side. It will look like this: X = 3 – 2 Then we simply subtract: x = 1

You will notice that we got the same answer as last time You will notice that we got the same answer as last time. Use whatever method makes the most sense to you and is easiest for you to remember. Try one on your own: X – 7 = 3

Did you get x = 10? Well done! Your solution should look like this: X – 7 = 3 X – 7 + 7 = 3 + 7 X = 10 Or like this: X = 3 + 7

Solving Algebraic Equations Using Multiplication and Division When we have an equation that has multiplication or division in it, we still want to isolate the variable by using the opposite operation. For example, 2x = 8 ***In algebra, if there is no operation between a number and a variable, it always means multiplication. So . . . Means 2 times x is equal to 8.

To isolate x, we are going to use division To isolate x, we are going to use division . 2x = 8 We need to “get rid of” the 2. If we divide the left side by 2, we are left with just x since 2 divided by 2 is 1. Since we have to do the same thing to the right side in order to keep our equation balanced, our solution would look like this: 2 2 X = 4

Here is another example: x = 7 2 In this question, we need to use multiplication to help us solve for x. Again, we need to “get rid of” the 2 in order to isolate x. Let’s multiply both sides by 2. 2 (x )= 7 (2) x = 14 **Since 2 over 2 equals 1, we are left with just x on the left side.

Check: x = 7 2 14 = 7 7 = 7 L.S. = R. S. Try a couple on your own: 3x = 18 x = 5 4

Your answers should look like this: 3x = 18 x = 5 4 3x = 18 4(x) = 5 (4) 3 3 4 x = 6 x = 20 GOOD JOB!!!

Solving Equations in Two+ Steps Some algebraic equations will involve more than one step in order to solve them. In this case, we use the same methods as before. . .we just combine them! e.g. 5x – 2 = 13 We still want to isolate x. First, we add 2 to each side and then we divide each side by 5. It should look like this:

5x – 2 = 13 5x – 2 + 2 = 13 + 2 5x = 15 5 5 x = 3 Check: 5 (3) – 2 = 13 15 – 2 = 13 13 = 13 LS = RS

Some questions may have variables and numbers on both sides of the equation. In this case, you have to collect “like terms.” We still want the variables on one side of the equation and the numbers on the other. e.g. 3x - 5 = 2x + 2 “Like terms” are terms that have the same variable and can be combined together. For example, in the above question, we can combine 3x and 2x because they both contain an x. Our solution would look like this: 3x - 5 + 5 – 2x = 2x + 2 – 2x + 5 ***we add 5 to both sides and subtract 2x from both sides 3x – 2x = 2 + 5 x = 7 Check: 3(7) – 5 = 2(7) + 2 21 – 5 = 14 + 2 16 = 16 LS = RS

Don’t forget . . . You can solve equations by inspection, guess and check, or using the balance method. When solving for a variable, always remember to isolate the variable (get it all by itself on one side of the equal sign) and to BALANCE the equation (whatever you do to one side of the equation, you must do to the other!!!). GOOD LUCK!!!