ACT Math : Algebra
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Example Questions
Example Question #3 : Systems Of Equations
John's age is half of Bob's age in 2010. John is also 3 years younger than Bob in 2010.
How old is Bob in 2013?
6
3
9
10
5
9
Using algebra, we can write the system of equations J = (1/2)B and J + 3 = B, where J is John's age in 2010 and B is Bob's age in 2010.
We plug in and get that J = 3 and B = 6
So 3 years later in 2013, we know Bob is 9 years old.
Example Question #4 : Systems Of Equations
Given the system of equations:
2x + 3y = 15
x + 3y = 0
What is x?
15
–5
–15
5
0
15
Substitute the second equation into the first to find that y = –5. Then plug y = –5 into either equation to find x = 15.
Example Question #11 : Systems Of Equations
Solve the system of equations:
2x + 3y = 10
x – 2y = –2
x = 0, y = 1
x = 2, y = 2
x = 0. y = 2
x = 2, y = 1
no solution
x = 2, y = 2
Use elimination to solve for one variable.
y = 2
Back substitute for the other variable.
x = 2
Example Question #1991 : Act Math
Given the system of equations:
2x + 2y = 12
x – y = 2
What is x equal to?
2
–2
8
4
1
4
Substitute the second equation into the first or use elimination.
Example Question #1992 : Act Math
Find a solution to the system of equations:
2x – y = 0
x + y = 3
(1,0)
(1,2)
(2,1)
(0,2)
(0,0)
(1,2)
Use substitution and plug in to solve for one equation. Then use back substitution to solve for the other variable.
Example Question #14 : How To Find The Solution For A System Of Equations
Tom's allowance is 3 times as much as Joe's allowance. If the sum of their allowance is $56, how much is Joe's allowance?
Tom's allowance (
The sum of their allowance is $56.
Restate the second equation by substituting 
Example Question #11 : How To Find The Solution For A System Of Equations
Solve
What is the sum of 
Add the two equations to get
or
and then substitute 
Solving for 
And we then substitute 
So the sum of 
Example Question #16 : How To Find The Solution For A System Of Equations
Jacob counts his money as he puts it in his new piggy bank. He has the same number of dimes as quarters and twice as many nickels as dimes. He has $2.25 total. How many dimes does he have?
The general equation to use is:
where 
Let 
So the equation to solve becomes
Solving shows that Jacob had 10 nickels, 5 dimes and 5 quarters for a total of $2.25.
Example Question #1994 : Act Math
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?
In 2010 the family sold 
This increased by 100 in 2011: 
In 2012 the number of apples tripled:
In 2013 the number of apples increased by 200:
If 
so in 2011 the number of apples sold was 500, or 
Example Question #11 : How To Find The Solution For A System Of Equations
If
and
Which of the following expresses 
First we must solve for 
Now that we have 
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 
This is much better as we can multiply straight across to get:
Now we can solve for 
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