1. Question 6

2. Question 6 Cont. 7x + 21 = 21 ______________ -21 -21 One Solution
__ 7
__ 7
To solve this one, you want to get the x variable on a side by itself. In order to that, you need to get the 21 to the other side of the equation. In this case, 21 is being added to the 7x so you want to do the opposite, which is subtract. As you see, in the next step, you subtract 21 from both sides. When you do that, the 21s cancel on both sides of the equation and you are left with 7x = 0. Since 7 and x are multiplied together, you need to do the opposite of that to get x by itself. The opposite of multiply is divide, so you will divide both sides by 7. You end up getting that x is equal to 0.
A lot of students will get confused when you subtract 21 from both sides and it cancels out both sets of 21. Students might choose no solution at that point because they get confused with no solution, since they forget that 0 can be an answer.
x = 0

3. Question 6 Cont b) 12x + 15 = 12x – 15 -15 -15 __________________
No Solution
12x = 12x – 30
-12x
-12x
________________
This solution ends up being No Solution.
The way I solved the problem is by the following steps:
First, I wanted to remove all of the numbers from the left side and place them on the right side of the equation. To do that, I subtracted 15 from both sides. When that happens, the 15s on the left hand side of the equation cancel out. On the right hand side of the equation, you have , which gives you -30.
Second, I wanted to get all of the x’s on the same side of the equation, meaning I wanted to move the ones on the right hand side to the left hand side. In order to do that, I subtracted 12x from both sides. When you subtract 12x from both sides, the 12x’s on the right hand side cancel out and you’re left with On the left hand side, both 12x cancel out, leaving you with 0 = -30
Since 0 is not the same as -30, there is no solution to this problem.
There’s another way to solve the problem without using math and looking at the problem. I’m not sure a lot of students will use this, but I wanted to share, just in case. The students will first examine the two equations. They are both linear equations (meaning they graph a straight line). Since the number in front of x is the same for both equations, it means they have the same slope. The other number (positive 15 and negative 15) represent where the line crosses the y-axis. Since both problems have the same slope, but different y-intercept, the two lines will never cross because they are parallel lines. This would show no solution as well.
0 = -30

4. Question 6 Cont c) -5x – 25 = 5x + 25 _______________ + 25 + 25
One Solution
-5x = 5x + 50
-5x
-5x
_______________
The solution to this problem is also one solution. The students might get confused because the numbers on both sides of the equation are opposite of each other, but if they just solved for x, they would see the solution.
In order to solve for the x, the students first need to move the -25 from the left hand side and put it on the right hand side. You do this by adding 25 to both sides. After adding 25 to both sides, the 25’s on the left hand side cancel out and you are left with -5x. The 25’s on the right hand side get added together and you now have 5x + 50 on that side.
Next, you want to get all of the x’s on the same side of the equation. This means you will subtract 5x from the right hand side (to cancel it out) and to the left hand side, because whatever you do to one side, you have to do to the other side. When you do this step, the 5x’s on the right hand side cancel out and you now have -10x on the left hand side.
In order to get x by itself, you need to divide by the number in front of x, which is You divide both sides by -10, and you get that x = -5. Since x is equal to a number, there is one solution to the problem.
-10x = 50
___ -10
___ -10
x = -5

5. Infinitely Many Solutions
Question 6
For each linear equation in this table, indicate whether the equation has no solution, one solution, or infinitely many solutions
Equation
No Solution
One Solution
Infinitely Many Solutions
7x + 21 = 21
12x + 15 = 12x – 15
-5x – 25 = 5x + 25
On this first slide, I am providing the answers. On each sequential slide, I am showing how to solve the problem to arrive at the answer.
Some quick definitions:
No Solution: The two sides of the equation do not equal each other when you solve for a variable. Example 0 = 30. Since 0 is not equal to 30, it does not work and there is no solution.
One Solution: The variable will be on one side by itself and it will be equal to a number. Example: x = 7. Infinitely Many Solutions: Both sides of the equation will equal each other. Example: 6 = 6. This one is not in this problem, however.
Calculators should be used on this problem. Also, in order to get credit for this problem, all parts need to be answered correctly.