ACT Math : Linear / Rational / Variable Equations
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Example Questions
Example Question #1790 : Act Math
Susan has one gallon of 25% acid solution, and she needs 75% acid solution for her experiment. How much pure acid should she add?
Pure water is 0% and pure acid is 100%.
The general formula to use is 
Our known values are listed below.
We will call our unknown added acid 
So the equation to solve becomes 
Example Question #291 : Algebra
Sally sells hair accessories. Her monthly fixed costs are $150. It costs her $5 to make each barrette, and she sells them for $10 each.
How many barrettes must Sally sell in order to break even with her costs?
The break-even point occurs when the revenues are the same as the costs.
Let 
Revenue is equal to 
Cost is equal to 
The equation to solve becomes
Solving shows that 30 barrettes must be sold to break-even.
Example Question #21 : How To Find The Solution To An Equation
Sally sells hair accessories. Her monthly fixed costs are $150. It costs her $5 to make each barrette, and she sells them for $10 each.
If Sally wants to make a profit of $250, how many barrettes must she sell?
Let 
Revenue will be equal to 
Costs will be equal to 
The equation to solve becomes 
To reach the profit goal of $250, we need to calculate the value of 
Example Question #22 : How To Find The Solution To An Equation
What are the values for x that satisfy the equation (x-d)(x+c)=0?
-dc
d and c
dc
–d and c
d and -c
d and -c
Set each component of this equation equal to zero and solve for d and c.
(x-d)(x+c)=0
x-d=0
d=0
x+c=0
x=-c
Example Question #21 : Linear / Rational / Variable Equations
If A + B = C and B – C = 3, what does A equal?
3
9
6
–9
–3
–3
This question can be solved using substitution. There are several ways to subsitute, but one possible method is to substitue (A+B) for C in the equation B-C=3: B-(A+B)=3. This simplifies to (B-A-B=3) (don't forget to distribute the negative!). Since (B-B)=0, -A must equal 3 (-A=3). If we divide both sides by -1, A=-3. If you got an answer of positive 3, you may have forgotten to divide both sides by -1.
Example Question #22 : Linear / Rational / Variable Equations
The population of a bird species is modeled by the following equation:
where 
12.2
20.4
280.8
38.5
23.0
20.4
Plugging in 130 for P, the equation becomes 130 = (11/8)x + 102. Solving for x, we obtain x = 20.4 years.
Example Question #25 : How To Find The Solution To An Equation
What is the value of c when 3c + 4 = 2c – 7?
–11
0
1
No Solution
11
–11
First put the variables on one side by subtracting 2c on both sides to get c + 4 = –7. Then subtract the 4 on both sides to arrive at your answer, c = –11.
Example Question #22 : How To Find The Solution To An Equation
If y=0.35(300 – 2y), what is the value of y to the nearest tenth?
61.8
51.2
62.3
65.0
61.8
Distributing the 0.35 to both the 300 and –2y leaves y=105 – 0.7y
Adding 0.7y to both sides and dividing by 1.7 gives 61.8.
Example Question #23 : Equations / Inequalities
Which of the lines below is not parallel to 4x – 3y = 17 ?
2x – 1.5y = 8.5
–y= –(4/3)x – (17/3)
8x + 6y = 34
8x – 6y = 34
8x + 6y = 34
For a line to be parallel with another, the slopes must be equal. All of the equations have a slope of 4/3 except 8x +6y=34.
Example Question #28 : How To Find The Solution To An Equation
If the expression x4 + 3cx – 2 is equal to 5 when x = –1, what is the value of c?
3
1
2
–2
–2
Plugging in –1 into the equation for x and solving for c yields –2.
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