>> Welcome to the Cypress College math review on solving simple linear equations.
Objective one.
Solving linear equations using properties of equality.
So we're trying to solve an equation.
So our goal is to get x on one side, isolate it, and simplify the other side.
We're going to use the addition
and multiplication properties of equality to help us do this.
Let's say that a, b and c are real numbers.
This property a plus c equals b plus c is equivalent
to a equals b is called the addition property of equality.
This property tells us that you're allowed to add
or subtract the same quantity to both sides of an equation.
ac equals bc is equivalent to ab is the multiplication property of equality.
This tells us that you can multiply or divide the same quantity to both sides of an equation.
We will use these two properties in solving equations.
Example. Solve for x. x minus 7 equals 3.
So to solve for x, we need to get x by itself, so we need to get rid of this minus 7 deal.
What do we do?
Well, we use the addition property of equality and we add 7.
Well, if you do it to one side, you must do the exact same thing to the other side.
So we need to add 7 to both sides of the equation.
Notice that minus 7 and plus 7 add up to 0 so that disappears and 3 plus 7 is equal to 10.
So on the left-hand side, all we have left is x. And on the right-hand side we have 10.
Therefore, x equals 10 and it's solved.
Example. Solve for x. 5 plus x equals 1.
Well, we wish to get x by itself, so we need to get rid of the 5.
That's a positive 5 out there and that's a separate term
from the x so we would need to subtract 5.
If we do that to the left side, we must do it to the right side.
So we subtract 5 from both sides.
A positive 5 and a negative 5 add up to 0.
So the only thing left on the left-hand side is x. 1 minus 5 is negative 4.
So x is equal to negative 4, and we have our solution.
Example. Solve for x. x divided by 8 equals 3.
So we wish to isolate the x. So we need to get rid of 8.
x is being divided by 8.
So the inverse operation to dividing is multiply.
So we need to use the multiplication property of equality and multiply both sides by 8.
A lot of students, when they're multiplying fractions, like to write the 8 as 8 over 1.
So now you see that the 8s cancel.
So we get x on the left-hand side.
x over 1 is just x. And then on the right-hand side we have 8 times 3 and that gives us 24.
So x is equal to 24.
5 times x equals negative 20.
So 5 is being multiplied times the x this time.
The inverse operation to multiply is divide.
So we divide both sides by 5.
Notice that the 5s cancel on the left-hand side so we just get x. x is isolated.
On the right-hand side, we have negative 20 divided by 5 and that's negative 4.
Therefore, x is equal to negative 4.
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Objective two.
Solving multi-step linear equations.
Example. Solve for y. The variable is not always going to be x.
So our goal is to get y isolated by itself.
We have negative 10 times y plus 15 equals 45.
Think of your order of operations.
You would multiply before you add, correct?
But to solve an equation, you reverse your order of operations to isolate the variable.
So we will subtract 15 before we divide by the negative 10.
So first subtract 15 from both sides.
Notice that positive 15 and negative 15 add up to 0.
So our left-hand side is just negative 10y.
On the right-hand side, we have 45 minus 15 and that's 30.
Now we have negative 10 times y equals 30.
The inverse operation to multiply is divide which means we need
to divide both sides by negative 10.
The negative 10s on the left-hand side cancel which just gives you 1y or just y.
On the right-hand side, 30 divided by negative 10 is negative 3.
Therefore, y is equal to negative 3.
Example. Solve for x. x divided by negative 2 plus 8 is equal to 11.
We wish to isolate x. We reverse our order of operations,
and we need to get rid of the plus 8 first.
Since it's plus 8, we would subtract 8.
So we subtract 8 from both sides.
Positive 8 and negative 8 add up to 0.
So we just have x divided by negative 2 on the left-hand side.
And 11 minus 8 is equal to 3.
Now we have x divided by negative 2.
We wish to get rid of the negative 2.
Since x is being divided by negative 2, the inverse operation to divide is multiply.
So we multiply by negative 2.
What we do to one side we have to do to the other side
so we multiply both sides by negative 2.
Since we're multiplying fractions, we'll go ahead
and write the negative 2 as negative 2 over 1.
And you can see that the negative 2s cancel.
So we just get 1x equals negative 2 times 3 which is negative 6 so x equals negative 6.
Example. Solve for x. 3x plus 7 equals 5x plus 7.
In this equation, we have variables on both sides and numbers on both sides.
So we need to get all of the x's on one side and all of the numbers on the other side.
So first let's move the x's.
I choose to move the x's to the right side.
So I'm going to move the 3x.
It doesn't matter.
You can choose either side.
So we would subtract 3x from both sides because that's a positive 3x.
When I subtract 3x from positive 3x, I get 0.
So I have 0 plus 7 equals.
On the right-hand side, I have 5x minus 3x which is 2x and I still have the plus 7.
Now I need to move the 7 to the left-hand side because I want to get x by itself.
So I subtract 7 from both sides.
On the left-hand side, 7 minus 7 is 0.
On the right-hand side, I have also 7 minus 7 is 0.
So I have 2x plus 0 which is just 2x.
I still do not have x by itself.
There is a 2 that's being multiplied times the x so I need
to do the inverse operation which is divide.
So I divide both sides by 2.
0 divided by 2 is 0.
And on the right-hand side the 2s cancel so I get 1x.
Well, that's just x so x is equal to 0.
And you can turn it around and put the x on the other side.
Example. Solve for x. 5x minus 8 equals 6x minus 1.
Once again we have variables on both sides and numbers on both sides, so we have to choose
which side we want to put the x's on.
I choose the left-hand side this time.
So I'm going to move the 6x to the left-hand side.
Since it's a positive 6x, I'm going to subtract 6x.
I subtract 6x from both sides.
5x minus 6x is negative 1x.
So on the left-hand side I have negative x minus 8.
On the right-hand side, 6x minus 6x is 0 and 0 minus 1 is negative 1.
Now we reverse our order of operations to isolate the x.
So since we have subtract 8, we need to add 8.
Add 8 to both sides.
Subtract 8 and add 8 gives us 0 so we have negative x plus 0 on the left-hand side.
On the right-hand side, we have negative 1 plus 8 and that's 7 so negative x is equal to 7.
We still do not have x isolated.
We need to get rid of the negative and that's the same thing as a negative 1.
So let's divide both sides by negative 1.
A negative divided by a negative is a positive so we get 1x on the left-hand side.
On the right-hand side, 7 divided by negative 1 is negative 7
so we get that x is equal to negative 7.
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Objective three.
Solving linear equations that can be simplified by combining like terms.
Example. Solve for x. 1 plus 4x plus 2 is equal to negative 3 plus 2x plus 14.
So we need to simplify each side of the equation before we start moving things around.
So on the left-hand side of the equation we have the like terms of 1 and 2.
So 1 plus 2 is equal to 3.
So on the left-hand side we have 3 plus 4x.
On the right-hand side, we have negative 3 plus 14.
Those are like terms.
We can add those up and get 11.
So the right-hand side is 11 plus 2x.
Now we need to move all of the x's to one side and all the numbers to the other side.
x's, I choose left side this time.
So we're going to move the 2x to the left-hand side
which means we need to subtract 2x from both sides.
4x minus 2x is equal to positive 2x.
On the left-hand side, we have 3 plus 2x.
On the right-hand side, 2x minus 2x is 0 and 11 plus 0 is just 11.
Now we reverse our order of operations to isolate the x.
So we subtract 3 first before we divide by the 2.
So we subtract 3 from both sides of the equation.
3 minus 3 is equal to 0 and we get 0 plus 2x which is just 2x.
On the right-hand side, we have 11 minus 3 and that's 8.
Now we have 2 times x is equal to 8.
We need to isolate the x. The inverse operation to multiply is divide.
So we divide both sides by 2.
See how the 2s cancel, and on the left-hand side I get 1x.
So we get that x is equal to 8 divided by 2 which is 4.
Example. Solve for x. Negative 2 times the quantity 3x minus 4 is equal to 2x.
Any time you have parentheses in an equation,
you must get rid of the parentheses before you start moving things around.
So we will distribute the negative 2 to the 3x which gives us negative 6x.
We distribute the negative 2 to the negative 4.
That gives us positive 8.
So the left-hand side is negative 6x plus 8.
The right-hand side is just 2x.
Now you can move the variables to the left or to the right
but since the right-hand side only has the variable 2x, it would be quicker to move all
of the x's to the right-hand side.
So I'll add 6x to both sides because I had a negative 6x.
Negative 6x plus 6x is equal to 0 so the left-hand side is just 8.
2x plus 6x is 8x.
8 is being multiplied times x so the next step is to divide both sides by 8.
The 8s cancel.
On the left-hand side I get 1 and the right-hand side I get 1x
which is just x. You can write the x in the answer either way on the left or the right
so 1 equals x is the same as x equals 1.
Example. Solve for k. The opposite
of the quantity 6k minus 5 minus the quantity negative 5k plus 8 is equal to negative 3.
So since we have parentheses, our first goal is to get rid of the parentheses.
A negative in front of a parentheses is the same thing as negative 1 times the parentheses.
So we're going to distribute a negative 1 to each of these parentheses.
The 6k inside the first parentheses will become a negative 6k when we multiply it by negative 1.
Subtract 5.
Well, that will become add 5 when we distribute the negative 1.
And negative 1 times negative 5k gives us positive 5k.
Negative 1 times positive 8 gives us negative 8.
And now we have the parentheses all gone.
Now we're going to have like terms.
On the left-hand side, we have a negative 6k and a 5k which adds to a negative 1k.
And we also have 5 minus 8.
Those add up to negative 3.
Now we wish to get k by itself.
So the first step would be to add 3 to both sides because we have a minus 3.
Minus 3 plus 3 is equal to 0.
So we get negative k on the left-hand side.
On the right-hand side, negative 3 plus 3 is equal to 0.
We still do not have the k isolated.
We need to get rid of the negative.
That's the same thing as negative 1k.
So we're going to divide both sides by negative 1.
Negative k divided by negative 1.
Negative divided by a negative is a positive so we just get k on the left-hand side.
0 divided by negative 1 is 0.
Therefore, k is equal to 0.
Example. Solve for m. Negative 3 times the quantity m minus 4 plus 2 times the quantity 5
plus 2m is equal to 29.
So our first goal is to get rid of the parentheses.
We distribute the negative 3 to m and get negative 3m.
Negative 3 times negative 4 gives us plus 12.
2 times 5 gives us 10.
And 2 times 2m gives us 4m.
Now we need to add like terms on the left-hand side of the equation.
Negative 3m and a positive 4m gives us positive 1m.
Positive 12 and positive 10 give us 22.
Now we have m plus 22 equals 29.
To get the m by itself, we need to subtract 22 from both sides.
So we get m equals 29 minus 22 which is 7 so m is equal to 7.
Pause the video and try these problems.