Solving Algebraic Equations One step equations with constants

Warm-Up: Complete the following questions and then click to check your answers. -6 - 11 = -17 (-1) - (-9) = 4 - (-2) = 6 (+3) + (-13) = -10 (-12) + (-5) = -17 -3 + 8 = 5 2– 5= 8 -3 (-3) + (+6) = 3

Our goal: To solve equations using algebra. You have already seen solving equations using inspection. For example, n+2=5 We know that n means “a number”. So, a number plus 2 equals 5 means that n = 3, since 3 + 2 = 5 Today, we are going to learn how to solve it algebraically, so that when the equations become more difficult, we will be able to solve them efficiently.

First of all, we need to learn how to represent equations using algebra tiles. We will use the following shapes to help us: n (the variable) +1 -1

Let’s represent some algebraic expressions using tiles. +3 -4 d m+2

Some more. n-1 3 f 2 w + 3 3 p - 1

Now, we are ready for equations. • Remember that equations contain an equal sign and two sides. • Also, equations are balances. Both sides are equal. • In order to keep our equations balanced, we need to do the same operations to both sides. • To solve an equation, we need to isolate the variable on one side of the equation.

Let’s try. We will represent the equation n – 3 = 6, using algebra tiles. Our goal is to isolate the variable, in this remove case n, on one side of the balance. Since there is also -3 on that side, we need to add +3, in order to create zero pairs. Remember: In order to maintain balance, we need to do the same thing to each side. n– 3=6 n– 3+3=6+3 n=9 Now, we can show our work algebraically, by writing down what we did. Next, remove any zero pairs and write down what’s left. Note that there are no zero pairs on the right side of the balance. n is now alone (isolated), so we are finished. We can see that n = 9. Let’s check our work on the next slide.

In order to check our work, we substitute n = 9 into the equation and check to make sure that both sides are equal. Substitute n = 9 Left side of equation n– 3 9– 3 6 Right side of equation 6 We can see that both sides are equal which means that our answer, n = 9, was correct.

Another. We will represent the equation -4= n + 1, using algebra tiles. Our goal is to isolate the variable, in this case n, on one side of the balance. Since there is also +1 on that side, we need to add -1, in order to create zero pairs. remove -4 = n + 1 -4 - 1 = n + 1 -5 = n Remember: In order to maintain balance, we need to do the same thing to each side. Now, we can show our work algebraically, by writing down what we did. Next, remove any zero pairs and write down what’s left. Note that there are no zero pairs on the right side of the balance. n is now alone (isolated), so we are finished. We can see that n = -5. Let’s check our work on the next slide.

In order to check our work, we substitute n = -5 into the equation and check to make sure that both sides are equal. Left side of equation -4 Substitute n = -5 We can see that both sides are equal which means that our answer, n = -5, was correct. Right side of equation n+1 -5 + 1 -4 If the sides are not equal, you either made a mistake when solving or when substituting.

Another. We will represent the equation n - 2 = -3, using algebra tiles. Our goal is to isolate the variable, in this remove case n, on one side of the balance. Since there is also -2 on the same side as n, we need to add +2, in order to create zero pairs. Remember: In order to maintain balance, we need to do the same thing to each side. n - 2 = -3 n - 2 + 2 = -3 + 2 n = -1 Now, we can show our work algebraically, by writing down what we did. Next, remove any zero pairs and write down what’s left. Note this time there are zero pairs on both sides of the balance. n is now alone (isolated), so we are finished. We can see that n = -1. Let’s check our work on the next slide.

In order to check our work, we substitute n = -1 into the equation and check to make sure that both sides are equal. Left side of equation n-2 Substitute n = -1 Right side of equation -3 -1 - 2 -3 We can see that both sides are equal which means that our answer, n = -1, was correct.

Another. We will represent the equation 1 = n + 3, using algebra tiles. Our goal is to isolate the variable, in this remove 1=n+3 1 -3=n+3 -3 -2 = n remove case n, on one side of the balance. Since there is also +3 on the same side as n, we need to add -3, in order to create zero pairs. Remember: In order to maintain balance, we need to do the same thing to each side. Now, we can show our work algebraically, by writing down what we did. Next, remove any zero pairs and write down what’s left. n is now alone (isolated), so we are finished. We can see that n = -2. Let’s check our work on the next slide.

In order to check our work, we substitute n = -2 into the equation and check to make sure that both sides are equal. Left side of equation 1 Substitute n = -2 Right side of equation n+3 -2 + 3 1 We can see that both sides are equal which means that our answer, n = -1, was correct.

Let’s practice. Try to save the following equation algebraically. Draw algebra tiles if needed. Make sure that you verify your answer. Click to check your work. Right Side Left Side Solve: n+5=2 2 n+5 -5=2 -5 n = -3 -3 + 5 2

Let’s practice. Try to save the following equation algebraically. Draw algebra tiles if needed. Make sure that you verify your answer. Click to check your work. Right Side Left Side Solve: -4 = n + 2 n+2 -4 -4 - 2 = n + 2 -6 = n -6 + 2 -4

Let’s practice. Try to save the following equation algebraically. Draw algebra tiles if needed. Make sure that you verify your answer. Click to check your work. Right Side Left Side Solve: n-6=7 7 n-6+6=7+6 n = 13 13 - 6 7

Let’s practice. Try to save the following equation algebraically. Draw algebra tiles if needed. Make sure that you verify your answer. Click to check your work. Right Side Left Side Solve: -1 = n - 8 -1 n-8 -1 + 8 = n - 8 + 8 7=n 7 -8 -1