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Algebra 1 : Equations / Solution Sets

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Example Question #11 : How To Find A Solution Set

Solve the following by substitution:

3x+y=2

y=x-2

Possible Answers:

x=-1, y=1

x=1, y=-1

No solution

x=1, y=1

x=-1, y=-1

Correct answer:

x=1, y=-1

Explanation:

To solve this equation you will plug the second equation straight into the first one by substituting what is written there for .

You will then get:

3x+(x-2)=2

From here you need to simplify by combining like terms:

4x-2=2

Bring the over by addition:

4x=4

Then divide both sides by to get:

x=1

You will then take this value of and plug it into either equation.

y=(1)-2=-1

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Example Question #12 : How To Find A Solution Set

Set A is composed of all multiples of 4 that are that are less than the square of 7. Set B includes all multiples of 6 that are greater than 0. How many numbers are found in both set A and set B?

Possible Answers:

Correct answer:

Explanation:

Start by making a list of the multiples of 4 that are smaller than the square of 7. When 7 is squared, it equals 49; thus, we can compose the following list:

$\text{4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, and 48}$

Next, make a list of all the multiples of 6 that are greater than 0. Since we are looking for shared multiples, stop after 48 because numbers greater than 48 will not be included in set A. The biggest multiple of 4 smaller that is less than 49 is 48; therefore, do not calculate multiples of 6 greater than 48.

$\text{6, 12, 18, 24, 30, 36, 42, and 48}$

Finally, count the number of multiples found in both sets. Both sets include the following numbers:

$\text{12, 24, 36, and 48}$

The correct answer is 4 numbers.

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Example Question #1 : How To Find The Solution For A System Of Equations

A cube has a volume of 216 $cm^{3}$ . If its width is $3x^{2}$ , its length is -0.5x , and its height is $x^{3}$ +14 , find .

Possible Answers:

Correct answer:

Explanation:

Since the object in question is a cube, each of its sides must be the same length. Therefore, to get a volume of 216 $cm^{3}$ , each side must be equal to the cube root of 216 , which is cm.

We can then set each expression equal to .

The first expression (3x^2^ -6) can be solved by either or , but the other two expressions make it evident that the solution is x = -2 .

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Example Question #1 : How To Find The Solution For A System Of Equations

3x + y = 8

2x + 4y = 12

Solve the system for and .

Possible Answers:

x = -4, y = 20

x = 10, y = -2

x = 2, y = 0

x = 2, y = 2

x = 3, y = -1

Correct answer:

x = 2, y = 2

Explanation:

The most simple method for solving systems of equations is to transform one of the equations so it allows for the canceling out of a variable. In this case, we can multiply 3x + y = 8 by (-4) to get -12x - 4y = -32 .

Then, we can add 2x + 4y = 12 to this equation to yield -10x = -20 , so x = 2 .

We can plug that value into either of the original equations; for example, 3(2) )+ y = 8 .

So, y = 2 as well.

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Example Question #2 : How To Find The Solution For A System Of Equations

What is the solution to the following system of equations:

3x+5y = 15

3x+3y = 15

Possible Answers:

x = 5, y = 0

x=3,y=2

$x=\frac{1}{3},y=3$

$x=2,y=\frac{13}{4}$

$x=3,y=\frac{6}{5}$

Correct answer:

x = 5, y = 0

Explanation:

By solving one equation for , and replacing in the other equation with that expression, you generate an equation of only 1 variable which can be readily solved.

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Example Question #1 : How To Find The Solution For A System Of Equations

Solve this system of equations for :

3x + 5y = 1

6x-y = 24

Possible Answers:

$2\frac{2}{3}$

$3\frac{2}{3}$

$3\frac{1}{3}$

None of the other choices are correct.

Correct answer:

$3\frac{2}{3}$

Explanation:

Multiply the bottom equation by 5, then add to the top equation:

6x-y = 24

$5 \left (6x-y \right ) =5\cdot 24$

30x-5y =120

$\underline{\textrm{\; } 3x + 5y = \; \; 1}$

$33x \;\;\;\;\;\; \; =121$

$x = \frac{121}{33} = \frac{11}{3} = 3 \frac{2}{3}$

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Example Question #2 : How To Find The Solution For A System Of Equations

Solve this system of equations for :

3x + 5y = 23

6x-y = -9

Possible Answers:

None of the other choices are correct.

$y = 3 \frac{1}{2}$

$y = 2 \frac{1}{2}$

y = 3

y = 5

Correct answer:

y = 5

Explanation:

Multiply the top equation by :

3x + 5y = 23

$-2 \cdot \left (3x + 5y \right )= -2 \cdot 23$

$-6x -10y \right )= -46$

Now add:

$6x-\; \; y = -9$

$\underline{-6x -10y = -46}$

-11y = - 55

$-11y \div (-11) = -55\div (-11)$

y = 5

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Example Question #3 : How To Find The Solution For A System Of Equations

Solve this system of equations for :

3x + 5y = 1

6x-y = 24

Possible Answers:

y = -2

$y = -1 \frac{3}{5}$

None of the other choices are correct.

$y = -2 \frac{1}{5}$

$y = -1 \frac{4}{5}$

Correct answer:

y = -2

Explanation:

Multiply the top equation by :

3x + 5y = 1

$-2 \cdot \left (3x + 5y \right )= -2 \cdot 1$

$-6x -10y \right )= -2$

Now add:

$\; \; 6x-\; \; \; y = 24$

$\underline{-6x -10y= -2} \right )$

-11y = 22

$-11y \div (-11) = 22\div (-11)$

y = -2

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Example Question #4 : How To Find The Solution For A System Of Equations

Find the solution to the following system of equations.

a + 3b = 5

a-2b = 0

Possible Answers:

$a = 1,\ b = 2$

$a = 1,\ b = 1$

$a = 2,\ b = 2$

$a = 2,\ b = 1$

$a = 2,\ b = 3$

Correct answer:

$a = 2,\ b = 1$

Explanation:

a + 3b = 5

a-2b = 0

To solve this system of equations, use substitution. First, convert the second equation to isolate $\small a$ .

$a-2b = 0\rightarrow a=2b$

Then, substitute $\small 2b$ into the first equation for $\small a$ .

a + 3b = 5

(2b) + 3b = 5

Combine terms and solve for $\small b$ .

5b=5

b=1

Now that we know the value of $\small b$ , we can solve for $\small a$ using our previous substitution equation.

a=2b=2(1)=2

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Example Question #1 : How To Find The Solution For A System Of Equations

Find a solution for the following system of equations:

$\left\{\begin{matrix} x-2y-1=3\\ -x+2y+2=5 \end{matrix}\right.$

Possible Answers:

$\left ( 4,0 \right )$

no solution

infinitely many solutions

$\left ( 8,2 \right )$

Correct answer:

no solution

Explanation:

When we add the two equations, the and variables cancel leaving us with:

1=8 which means there is no solution for this system.

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Algebra 1 : Equations / Solution Sets

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #11 : How To Find A Solution Set

Example Question #12 : How To Find A Solution Set

Example Question #1 : How To Find The Solution For A System Of Equations

Example Question #1 : How To Find The Solution For A System Of Equations

Example Question #2 : How To Find The Solution For A System Of Equations

Example Question #1 : How To Find The Solution For A System Of Equations

Example Question #2 : How To Find The Solution For A System Of Equations

Example Question #3 : How To Find The Solution For A System Of Equations

Example Question #4 : How To Find The Solution For A System Of Equations

Example Question #1 : How To Find The Solution For A System Of Equations

All Algebra 1 Resources

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