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High School Math : Solving Equations

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Example Question #1 : Equations

Cindy's Cotton Candy sells cotton candy by the bag. Her monthly fixed costs are $\$150$ . It costs $\$2.50$ to make each bag and she sells them for $\$4.00$ .

What is the monthly break-even point?

Possible Answers:

$225\; bags$

$60\; bags$

$100\; bags$

$150\; bags$

$80\; bags$

Correct answer:

$100\; bags$

Explanation:

Costs=150+2.50x

Revenues = 4.00x

The break-even point occurs when the costs=revenues .

The equation to solve becomes

4x=150+2.5x so the break-even point is $100 \; bags$ .

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Example Question #1 : Equations

Cindy's Cotton Candy sells cotton candy by the bag. Her monthly fixed costs are $\$150$ . It costs $\$2.50$ to make each bag and she sells them for $\$4.00$ .

To make a profit of $\$150$ , how many bags of cotton candy must be sold?

Possible Answers:

$250\; bags$

$200\; bags$

$100\; bags$

$300\; bags$

$150\; bags$

Correct answer:

$200\; bags$

Explanation:

Costs=150+2.50x

Revenues = 4.00x

Profits = Revenues - Costs = 4x - (150+2.5x)=1.5x -150

So the equation to solve becomes 150 = 1.5x - 150 , or $x=200\; bags$ must be sold to make a profit of $\$150$ .

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Example Question #3 : Equations

Solve for and to satisfy both equations in the system:

$\begin{Bmatrix} 6x+y = 25 \\ 2x-3y = 25 \end{Bmatrix}$

Possible Answers:

x=5 , y = 5

x = 5 , y= -5

x = -5 , y = 5

x= -5 , y = -5

x = 3 , y = 7

Correct answer:

x = 5 , y= -5

Explanation:

The two equations in this system can be combined by addition or subtraction to solve for and . Isolate the variable to solve for it by multiplying the top equation by so that when the equations are combined the term disappears.

$3\times(6x + y = 25)$

$\rightarrow 18x+3y = 75$

$\rightarrow (18x +3y = 75) + (2x - 3y = 25)$

$\rightarrow 20x = 100$

Divide both sides by to find as the value for .

Substituting for in both of the two equations in the system and solving for gives a value of for .

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Example Question #1 : Solving Equations

Solve for :

$\left | 12x - 39 \right | = 60$

Possible Answers:

$x = - 8 \frac{1}{4} \textrm{ or }x = 8 \frac{1}{4}$

$x = -1 \frac{3}{4} \textrm{ or }x = 8 \frac{1}{4}$

$x = 1 \frac{3}{4} \textrm{ or }x = 8 \frac{1}{4}$

$x = 8 \frac{1}{4}$

$x = 1 \frac{3}{4}$

Correct answer:

$x = -1 \frac{3}{4} \textrm{ or }x = 8 \frac{1}{4}$

Explanation:

Rewrite $\left | 12x - 39 \right | = 60$ as a compound statement and solve each part separately:

$\left | 12x - 39 \right | = 60$

$12x - 39 = - 60 \textrm{ or }12x - 39 = 60$

12x - 39 = - 60

12x - 39 + 39 = - 60+ 39

12x = - 21

$12x \div 12 = - 21 \div 12$

$x = -1 \frac{3}{4}$

12x - 39 = 60

12x - 39 + 39 = 60+ 39

12x = 99

$12x \div 12 = 99 \div 12$

$x = 8 \frac{1}{4}$

The solution set is $\left \{ -1 \frac{3}{4} ,8 \frac{1}{4}\right \}$

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Example Question #1 : Equations

a^2-b^2 = 33 and a-b=3 . What is the value of $a^{2}+b^{2}$ ?

Possible Answers:

130

112

Correct answer:

Explanation:

First, notice that we can factor a^2-b^2 into the form (a-b)(a+b). We are told that a-b=3, so we can substitute that into the first equation.

(a-b)(a+b)=33

3(a+b)=33

If we divide both sides by 3, we can obtain the value of a+b.

a+b=11

We now have a system of equations: a-b = 3, a+b = 11. We will solve this system by elimination. If we add the two equations together, we obtain the following:

2a=14 .

Divide both sides by 2.

a=7.

Going back to the equation a-b = 7, we can solve for b.

7-b=3

Add b to both sides.

7=3+b

Subtract 3 from both sides.

b=4.

Ultimately, the question asks us to determine the value of a^2+b^2 .

a^2+b^2 = $7^{2}+4^{2}=49+16=65$ .

The answer is 65.

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Example Question #2 : Solving Equations

Solve the pair of equations for x and y:

x+y=4

x-y=8

Possible Answers:

x=-6,y=2

x=-6,y=-2

x=6, y=-2

x=-2,y=6

x=2,y=-6

Correct answer:

x=6, y=-2

Explanation:

Equation 1: x+y=4

Equation 2: x-y=8

Solve equation 2 for X: x=8+y

Substitute into Equation 1: (8+y)+y=4

Solve for y: 2y=-4 , y=-2

Take the answer for y and plug it back into either original equation to find x:

x=6, y=-2

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Example Question #3 : Solving Equations

Solve the pair of equations for x and y:

2x+y=16

y-2x=0

Possible Answers:

x=8,y=-4

x=8,y=4

x=4,y=-8

x=-4,y=8

x=4,y=8

Correct answer:

x=4,y=8

Explanation:

Equation 1: 2x+y=16

Equation 2: y-2x=0

Solve Equation 2 for y:

y=2x

Substitute into Equation 1:

2x+2x=16

4x=16

x=4

Plug x back into either original equations and solve for y:

x=4,y=8

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Example Question #3 : Solving Equations

For which of the following functions is the result of f(-2) a positive integer?

Possible Answers:

2x-10

6x^3

2x^2-13

2x-5

5x^2 + 2

Correct answer:

5x^2 + 2

Explanation:

Simply plugging in -2 into each answer choice will determine the correct answer:

6x^3 = 6(-2)^3 = -48

2x-5=2(-2)-5=-9

2x^2-13=2(-2)^2-13=-5

5x^2 + 2 = 5(-2)^2 + 5 = 25

2x-10 = 2(-2) - 10 = -14

The key to solving this problem is to remember the order of operations and that negative numbers squared are positive, while negative numbers raised to the third power are negative.

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Example Question #4 : Solving Equations

Given the equation x^3 -6 = 210 , what is the value of 2x + 3 ?

Possible Answers:

Correct answer:

Explanation:

When solving the equation x^3 - 6 = 210 , observe that x^3 = 216 . Taking the cube root of 216 gives 6. Thus, 2x + 3=2(6) + 3=15.

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Example Question #5 : Solving Equations

A cardboard packing box contains footballs and baseballs. The ratio by weight of baseballs to footballs is 7 to 9. How many kilograms of footballs will there be in the box if the total weight of the box is 48 kilograms?

Possible Answers:

Correct answer:

Explanation:

In the box, there are kilograms of baseballs and kilograms of footballs. In total, there are 16x kilograms of balls. The total weight of the box is 48 kilograms, so $16x = 48\: and\: x = 3.$ Since there are kilograms of footballs, the total weight of the footballs in the box is equal to: 9(3) = 27.

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High School Math : Solving Equations

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Equations

Example Question #1 : Equations

Example Question #3 : Equations

Example Question #1 : Solving Equations

Example Question #1 : Equations

Example Question #2 : Solving Equations

Example Question #3 : Solving Equations

Example Question #3 : Solving Equations

Example Question #4 : Solving Equations

Example Question #5 : Solving Equations

All High School Math Resources

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