All ACT Math Resources
Example Questions
Example Question #511 : Algebra
Solve for x based on the following system of equations:
x + y = 5
2x + 3y = 20
10
–2
–5
–10
5
–5
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 #2002 : Act Math
If st + 12= 3s + sv, and t – v = 7, what is the value of s?
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 : Systems Of Equations
For what negative value of n does the system of equations yield no solutions?
3x + ny = 17
nx + (n + 6)y = 7
–5
–3
–1
–2
–3
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?
6
1
5
None of the answers are correct
9
1
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 #2004 : Act Math
How much pure water should be added to 1 gallon of pure cleaning solution to dilute it to 60% strength by volume?
None of the answers are correct
2/3 gallon
1/3 gallon
1/4 gallon
1/6 gallon
2/3 gallon
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?
12
9
10
11
14
10
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 #22 : Systems Of Equations
Given the following two equations, solve for :
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 #512 : Algebra
What value of and solve the following system of equations?
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 .
add to both sides.
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:
Example Question #24 : Systems Of Equations
Solve and . What is the sum of and ?
Adding the two equations together gives , so . Substituting into one of the original equations gives
The sum of and is
Example Question #242 : Equations / Inequalities
Suppose x2 + x – 6 = 0. Which of the following could be a value of x?
4
3
2
0
1
2
Factor out this binomial. –3 and 2 are the only possible x values. 2 is the answer.