What is the solution to the equation 5(x-6)=2(x+3)?
2 answers:
When solving equations in algebra, you isolate the variable you're solving for.
In this case, we want to solve for x.
And how would we solve for x? Well, we would want to get x on one side of the equation all alone and everything else to do the opposite side of x.
We get x all alone on one side by undoing all the operations done to it.
We undo operations by using the inverse of the operation.
You'll see what I mean when I start solving the equation.
If the variable you're solving for is on both sides of the equation, we will need to get them on one side.
Typically, it would be the left-hand side. But if you prefer the right-hand side that will also work.
We also have parentheses. We will need to simplify them using the distributive property.
The distributive property states
Now let's solve the equation.
5(x - 6) = 2(x + 3)
Simplify the parentheses on both sides using the distributive property.
5(x - 6) = 2(x + 3)
5x - 30 = 2x + 6
Great! We simplified the parentheses. But we have one x term on one side and another x term on the opposite side.
I want to get all the x terms on the left-hand side of the equation.
What we can do is subtract both sides by from equation by 2x.
And it is important to note that if you do something on one side, you do it on the opposite side.
5x - 30 - 2x = 2x + 6 - 2x
3x - 30 = 6
Now we just have an x term on one side of the equation.
Now solving it will be a lot easier.
The x term is being multiplied by 3 and is being subtracted by 30.
We will need to undo all those operations. But how?
We want to use the inverse of subtraction and multiplications.
Which are addition and division respectively.
First, undo the subtraction by adding 30 to both sides of the equation.
We do the same operation to both sides of an equation to keep the equation balanced or true.
3x - 30 + 30 = 6 + 30
3x = 36
We are very close to the answer. Just one more step.
The x term is being multiplied by 3.
We can undo the multiplication operation by dividing both sides by 3.
3x / 3 = 36 / 3
x = 12
So, x is equal to 12.
In this case, we want to solve for x.
And how would we solve for x? Well, we would want to get x on one side of the equation all alone and everything else to do the opposite side of x.
We get x all alone on one side by undoing all the operations done to it.
We undo operations by using the inverse of the operation.
You'll see what I mean when I start solving the equation.
If the variable you're solving for is on both sides of the equation, we will need to get them on one side.
Typically, it would be the left-hand side. But if you prefer the right-hand side that will also work.
We also have parentheses. We will need to simplify them using the distributive property.
The distributive property states
Now let's solve the equation.
5(x - 6) = 2(x + 3)
Simplify the parentheses on both sides using the distributive property.
5(x - 6) = 2(x + 3)
5x - 30 = 2x + 6
Great! We simplified the parentheses. But we have one x term on one side and another x term on the opposite side.
I want to get all the x terms on the left-hand side of the equation.
What we can do is subtract both sides by from equation by 2x.
And it is important to note that if you do something on one side, you do it on the opposite side.
5x - 30 - 2x = 2x + 6 - 2x
3x - 30 = 6
Now we just have an x term on one side of the equation.
Now solving it will be a lot easier.
The x term is being multiplied by 3 and is being subtracted by 30.
We will need to undo all those operations. But how?
We want to use the inverse of subtraction and multiplications.
Which are addition and division respectively.
First, undo the subtraction by adding 30 to both sides of the equation.
We do the same operation to both sides of an equation to keep the equation balanced or true.
3x - 30 + 30 = 6 + 30
3x = 36
We are very close to the answer. Just one more step.
The x term is being multiplied by 3.
We can undo the multiplication operation by dividing both sides by 3.
3x / 3 = 36 / 3
x = 12
So, x is equal to 12.
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Re-arranging the formula, we get
And substituting v=66 ft/s, we find
2. c. y = 5 - n
In fact, we can check that this rule satisfies all the numbers of the sequence:
n=3 --> y = 5 - 3 = 2
n=4 --> y = 5 - 4 = 1
n=5 --> y = 5 - 5 = 0
n=6 --> y = 5 - 6 = -1
3. d. $5.50, $11.00, $16.50; E = 5.50h
In fact, if we use the rule E=5.50 h, we see that the table is completed as follows:
h=1 --> E=(5.50)(1 h)=5.50$
h=2 --> E=(5.50)(2 h)=11.00 $
h=3 --> E=(5.50)(3 h)=16.50$
h=4 --> E=(5.50)(4 h)=22.00$
h=5 --> E=(5.50)(5 h)=27.50$
4. c. if d ≥ 5 then C = 4.70d, if d < 5 then C = 4.70d - 1; $17.80, $32.90
If the number of days is more than 5, the cost is 4.70$ per day, therefore c=4.70d (where d is the number of days); if the number of days is less than 5, one dollar is returned, so the cost will be C=4.70d-1. By applying this rule, we can calculate the rental cost after d=4 days and d=7 days:
- d=4 --> c=4.70*4-1=17.80$
- d=7 --> c=4.70*7=32.90$
5. b. y = 3x + 3
In fact, we can check this is the correct rule by substituting the numbers:
x=0 --> y=3*0+3=3
x=1 --> y=3*1+3=6
x=2 --> y=3*2+3=9
x=3 --> y=3*3+3=12
x=4 --> y=3*4+3=15
7. x -2 -1 0 1 2
y (-16 ) (-6 ) (4 ) (14 ) (24 )
The rule we have to apply is:
-10x + y = 4
Re-arranging it, we can solve for y:
y = 10x + 4
So, let's substitute the different values of x:
x=-2 --> 10*(-2)+4=-20+4= -16
x=-1 --> 10*(-1)+4=-10+4= -6
x=0 --> 10*0+4= 4
x=1 --> 10*1+4=10+4= 14
x=2 --> 10*2 +4 =20+4 = 24
8. y = -5x - 1
In fact, if substitute the different values of x given by the problem, we find:
x = -2 --> y = -5*(-2) -1= 10 -1 = 9
x = -1 --> y = -5*(-1) -1 = 5 -1 = 4
x = 0 --> y = -5*0 -1 = -1
x = 1 --> y = -5*1 - 1 = -6
x = 2 --> y = -5*2 -1 = -11