# ACT Math : Equations / Inequalities

## Example Questions

### Example Question #17 : How To Find The Solution For A System Of Equations

Jenna's family owns a fruit stand.  They began selling apples in 2010.  In 2011 the number of apples sold increased by 100 apples.  In 2012 they sold three times as many apples as they had in 2011, and in 2013 the number of apples increased by 200. If they sold 1700 apples in 2013, how many apples did they sell in 2011?

Explanation:

In 2010 the family sold number of apples.

This increased by 100 in 2011: .

In 2012 the number of apples tripled:

In 2013 the number of apples increased by 200:

If , we must solve for.

so in 2011 the number of apples sold was 500, or  apples.

### Example Question #18 : How To Find The Solution For A System Of Equations

If

and

Which of the following expresses  in terms of ?

Explanation:

First we must solve for , then substitute into the other equation. Since we want in terms of , solve for  in the equation and substitute our value of  (in terms of ) into the equation, then simplify:

Now that we have , let's plug that into the equation.

Already we can see that this problem is a mess because it is an expression with two denominators. Remember that dividing by a number is equal to multiplying by that number's inverse. Thus, dividing by  is the same as multiplying by . So let's make an equivalent expression look like this:

This is much better as we can multiply straight across to get:

Now we can solve for .

### Example Question #19 : How To Find The Solution For A System Of Equations

Solve for x based on the following system of equations:

x + y = 5

2x + 3y = 20

2

10

5

10

5

5

Explanation:

One method of solving a system of equations requires multiplying one equation by a factor that will allow for the removal of one variable. In this system, we can multiply (x+y=5) by -2. When the -2 is distributed across the entire equation, the equation becomes (-2x-2y=-10). We then add the two equations: (-2x-2y=-10) + (2x+3y=20). When we do this, the x variable cancels out, leaving us with y=10. We then subsitute 10 for y in either of the original equations: (x+10=5) or (2x+ 30=20). Either way, we end up with x=-5. If you got an answer of 5, you may have made a computation error. If you got 10, you may have forgotten to substitute the y-value into one of the original equations to solve for x.

### Example Question #20 : How To Find The Solution For A System Of Equations

If st + 12= 3s + sv, and t – v = 7, what is the value of s?

Explanation:

We are given st + 12= 3s + sv and t-v=7. Since we have a numerical value for t-v+, it would be ideal to isolate that in the first equation. We can do this be rearranging the first equation. Starting with st + 12= 3s + sv, we can subtract the 12 from the left giving us a -12 on the right. We then subtract the sv from the right and put it on the left. This gives us st-sv=3s-12. We then factor out the s on the left side of the equation, giving us s(t-v)=3s-12. We then plug-in 7 for t-v, giving us 7s=3s-12. We then solve for s, giving us -3.

### Example Question #21 : How To Find The Solution For A System Of Equations

For what negative value of n does the system of equations yield no solutions?

3x + ny = 17
nx + (n + 6)y = 7

2

3

1

5

3

Explanation:

Plugging in 3 for n gives a system that, when added vertically, gives 0 = 24, which is untrue.

### Example Question #22 : How To Find The Solution For A System Of Equations

Consider the following system of equations:  x – y = 5 and 2x + y = 4.

What is the sum of x and y?

9

1

6

None of the answers are correct

5

1

Explanation:

Add the two equations to get 3x = 9, so x = 3.  Substitute the value of x into one of the equations to find the value of y; therefore x = 3 and y = -2, so their sum is 1.

### Example Question #23 : How To Find The Solution For A System Of Equations

How much pure water should be added to 1 gallon of pure cleaning solution to dilute it to 60% strength by volume?

2/3 gallon

1/3 gallon

None of the answers are correct

1/4 gallon

1/6 gallon

2/3 gallon

Explanation:

Pure water is considered 0% whereas pure solution is 100%.

The general equations is Vwater x Pwater + Vsoultion x Psolution = Vfinal x Pfinal where

V means volume and P means percent.

x(0) + 1(1.00) = (x + 1)(0.60) and solve for x = volume of pure water.

### Example Question #24 : How To Find The Solution For A System Of Equations

Joey is four years older than Billy. The sum of their ages is 24. How old is Billy?

9

11

12

14

10

10

Explanation:

Define variables as x = Billy's age and x + 4 = Joey's age

The sum of their ages is x + (x + 4) = 24

Solving for x, we get that Billy is 10 years old and Joey is 14 years old.

### Example Question #25 : How To Find The Solution For A System Of Equations

Given the following two equations, solve for :

Explanation:

Solution A:

Notice that the two equations have very similar terms.  If the two expressions are subtracted from each other, the variable  cancels out:

(don't forget to distribute the minus sign throughout!)

-----------------------

Solution B:

Using one of the equations, solve for a in terms of b:

Note:  Solution A is the much faster way to solve this problem.  Whenever you are asked to solve a problem with two equations and two variables (or more!), see if you can add them together or subtract them from each other to make the other variables cancel out.

### Example Question #26 : How To Find The Solution For A System Of Equations

What value of  and  solve the following system of equations?

Explanation:

To solve the system of equations,  and , we must begin by inserting one equation into another. First notice that there are two variables and that we have two equations, therefore, we have enough equations to solve for both variables. As an aside, for each variable, you need that many equations to solve for all variables.

We will insert the equations by substitution, which tends to be a more useful form of equation integration. As the is the easier equation to solve for one variable, we’ll start there. Solve for

From here can insert this equation into all ’s in our other equation and solve

distribute

combine like terms

subtract 9 from both sides

divide by 6

From here we reinsert this value into our equation.

Solution: