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Algebra 1 : Linear / Rational / Variable Equations

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #2 : Solving Equations

Solve for :.

18x^3 - 8x = 0

Possible Answers:

$\left \{-\frac{2}{3}, \frac{2}{3} \right \}$

$\left \{ - \frac{3}{2}, \frac{3}{2} \right\}$

$\left \{ \frac{3}{2}, 0, -\frac{3}{2} \right\}$

$\left \{-\frac{2}{3}, 0, \frac{2}{3} \right \}$

Correct answer:

$\left \{-\frac{2}{3}, 0, \frac{2}{3} \right \}$

Explanation:

First factor the expression by pulling out :

18x^3 - 8x = 0

2x(9x^2-4) = 0

Factor the expression in parentheses by recognizing that it is a difference of squares:

2x(3x+2)(3x-2)= 0

Set each term equal to 0 and solve for the x values:

$2x=0 \rightarrow x=0$

$3x+2=0\rightarrow x=-\frac{2}{3}$

$3x-2=0\rightarrow x=\frac{2}{3}$

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Example Question #21 : Linear / Rational / Variable Equations

Possible Answers:

$\frac{4}{3}$

$\frac{5}{9}$

$\frac{5}{3}$

$\frac{7}{9}$

$\frac{2}{3}$

Correct answer:

$\frac{5}{9}$

Explanation:

1. First, simplify the numerator by finding a common denominator:

$\frac{6}{3}-\frac{4}{12}=\frac{6}{3}-\frac{1}{3}=\frac{5}{3}$

2. Next, simplify the denominator. These fractions can be added together without any additional work:

$\frac{9}{5}+\frac{6}{5}= \frac{15}{5}=3$

3. Then, simplify by multiplying top and bottom by the recriprocal of the denominator:

$\frac{\frac{5}{3}}{3}\cdot\frac{\frac{1}{3}}{\frac{1}{3}}=\frac{\frac{5}{9}}{1}=\frac{5}{9}$

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Example Question #22 : Linear / Rational / Variable Equations

$A\bigtriangledown B = \frac{-A^2+3(-B)^2}{\frac{2}{A}+\frac{1}B{}}$

$4\bigtriangledown 2=$

Possible Answers:

$-\frac{3}{2}$

$-\frac{1}{2}$

Correct answer:

Explanation:

Here, A=4,B=2 . Plug these values in and simplify.

Start with the numerator first:

-A^2 +3(-B)^2 = -(4)^2 +3(-2)^2= -16 +12=-4

Then do the same in the denominator:

$\frac{2}{A}+\frac{1}{B}= \frac{2}{4}+\frac{1}{2}= \frac{1}{2}+\frac{1}{2}=1$

Finally, combine the two results to find the solution:

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

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Example Question #23 : Linear / Rational / Variable Equations

Solve for :

$(x-3)^{2}=36$

Possible Answers:

(6,-6)

(9,-3)

(3,-9)

(9,3)

None of the other answers

Correct answer:

(9,-3)

Explanation:

To solve this equation, you must first eliminate the exponent from the (x-3) by taking the square root of both sides:

$\sqrt{(x-3)^{2}}=\sqrt{36}$

Since the square root of 36 could be either or , there must be 2 values of . So, solve for

x-3=-6

and

x-3=6

to get solutions of (9,-3) .

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Example Question #24 : Linear / Rational / Variable Equations

Solve for :

ax-b=cd

Possible Answers:

$\frac{a}{cd+b}$

None of the other answers

$\frac{cd+a}{b}$

$\frac{cd+b}{a}$

$\frac{b}{cd+a}$

Correct answer:

$\frac{cd+b}{a}$

Explanation:

To solve for , you must isolate it from the other variables. Start by adding to both sides to give you ax=cd+b . Now, you need only to divide from both sides to completely isolate . This gives you a solution of $x=\frac{cd+b}{a}$ .

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Example Question #2 : Solve Linear Equations With Rational Number Coefficients: Ccss.Math.Content.8.Ee.C.7b

Solve for :

6x-3-4x+5=3(x-1)

Possible Answers:

Correct answer:

Explanation:

Combine like terms on the left side of the equation: 2x+2=3(x-1)

Use the distributive property to simplify the right side of the equation: 2x+2=3x-3

Next, move the 's to one side and the integers to the other side: x=5

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Example Question #22 : Systems Of Equations

Solve for :

ex+f=g

Possible Answers:

$\frac{e}{g-f}$

$\frac{g+f}{e}$

$\frac{e}{g+f}$

$\frac{g-f}{e}$

None of the other answers

Correct answer:

$\frac{g-f}{e}$

Explanation:

To solve for , you must isolate it so that all of the other variables are on the other side of the equation. To do this, first subtract from both sides to get ex=g-f . Then, divide both sides by to get $x=\frac{g-f}{e}$ .

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Example Question #151 : Equations / Inequalities

The sum of four consecutive integers equal 98. What is the sum of the smallest and the largest number in the set?

Possible Answers:

Correct answer:

Explanation:

Let a variable represent the smallest number in the set. Every consecutive number after the first number will be one unit greater than the other. Then, the four numbers that sum up to 98 are:

(x)+(x+1)+(x+2)+(x+3)=98

where every term in parenthesis represents a number. Add and solve for .

4x+6=98

x=23

The smallest number is , and the largest number is x+3 . Therefore:

x+(x+3)= 23+(23+3)=23+26=49

The correct answer is:

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Example Question #152 : Equations / Inequalities

Solve for .

3a+4-4(a+2)=-12a+6(a-2)

Possible Answers:

-8/5

5/8

8/5

-5/8

Correct answer:

-8/5

Explanation:

The question asks for the value of .

To isolate , use the distributive properties to remove the parentheses then combine like terms.

3a+4-4(a+2)=-12a+6(a-2)

3a+4-4a-8=-12a+6a-12

-a-4=-6a-12

Finally isolate by balancing the equation.

5a=-8

a=-8/5

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Example Question #153 : Equations / Inequalities

Find the solution to this equation: $10x+25(x-\frac{1}{25})=174$

Possible Answers:

x=5

$x=\frac{1}{5}$

x=-5

None of the other answers.

$x=-\frac{1}{5}$

Correct answer:

x=5

Explanation:

To find the solution to this equation you must completely isolate x.

$10x+25(x-\frac{1}{25})=174$

Start by dealing with the parenthesis first. Distribute the 25:

10x+25x-1=174

Combine the x terms and move the constant over:

35x=175

Finally divide the constant away from x to solve:

x=5

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Algebra 1 : Linear / Rational / Variable Equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #2 : Solving Equations

Example Question #21 : Linear / Rational / Variable Equations

Example Question #22 : Linear / Rational / Variable Equations

Example Question #23 : Linear / Rational / Variable Equations

Example Question #24 : Linear / Rational / Variable Equations

Example Question #2 : Solve Linear Equations With Rational Number Coefficients: Ccss.Math.Content.8.Ee.C.7b

Example Question #22 : Systems Of Equations

Example Question #151 : Equations / Inequalities

Example Question #152 : Equations / Inequalities

Example Question #153 : Equations / Inequalities

All Algebra 1 Resources

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