Algebra II : Equations

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #271 : Solving Equations

Solve the equation:  \(\displaystyle \frac{x}{5}+\frac{5}{3} = 5\)

Possible Answers:

\(\displaystyle \frac{75}{4}\)

\(\displaystyle \frac{50}{3}\)

\(\displaystyle \frac{25}{3}\)

\(\displaystyle \frac{35}{3}\)

\(\displaystyle \frac{1}{5}\)

Correct answer:

\(\displaystyle \frac{50}{3}\)

Explanation:

In order to eliminate the fractions, we can multiply both sides by the least common denominator.

The least common denominator can be determined by multiplying the two denominators together.

\(\displaystyle 5\times 3 = 15\)

\(\displaystyle (15)(\frac{x}{5}+\frac{5}{3}) = 5(15)\)

The equation becomes:

\(\displaystyle 3x+25 = 75\)

Subtract both sides by 25.

\(\displaystyle 3x+25 -25= 75-25\)

\(\displaystyle 3x=50\)

Divide by three on both sides.

\(\displaystyle \frac{3x}{3}=\frac{50}{3}\)

The answer is:  \(\displaystyle \frac{50}{3}\)

Example Question #412 : Equations

Solve the equation:  \(\displaystyle 9-4x= 11-9x\)

Possible Answers:

\(\displaystyle \frac{20}{13}\)

\(\displaystyle 4\)

\(\displaystyle \frac{2}{13}\)

\(\displaystyle -\frac{2}{13}\)

\(\displaystyle \frac{2}{5}\)

Correct answer:

\(\displaystyle \frac{2}{5}\)

Explanation:

Add \(\displaystyle 9x\) on both sides.

\(\displaystyle 9-4x+9x= 11-9x+9x\)

Combine like-terms on both sides.

\(\displaystyle 9+5x =11\)

Subtract nine from both sides.

\(\displaystyle 9+5x -9=11-9\)

\(\displaystyle 5x=2\)

Divide by five on both sides.

\(\displaystyle \frac{5x}{5}=\frac{2}{5}\)

The answer is:  \(\displaystyle \frac{2}{5}\)

Example Question #413 : Equations

Solve the equation:  \(\displaystyle 8x-7 = 2x+7\)

Possible Answers:

\(\displaystyle \frac{7}{3}\)

\(\displaystyle \frac{3}{8}\)

\(\displaystyle -\frac{7}{3}\)

\(\displaystyle 0\)

\(\displaystyle \frac{14}3\)

Correct answer:

\(\displaystyle \frac{7}{3}\)

Explanation:

Subtract \(\displaystyle 2x\) from both sides.

\(\displaystyle 8x-7-2x = 2x+7-2x\)

The equation becomes: 

\(\displaystyle 6x-7=7\)

Add seven on both sides.

\(\displaystyle 6x-7+7=7+7\)

\(\displaystyle 6x=14\)

Divide by six on both sides and reduce the fraction.

\(\displaystyle \frac{6x}{6}=\frac{14}{6}\)

The answer is:  \(\displaystyle \frac{7}{3}\)

Example Question #414 : Equations

Solve the equation:  \(\displaystyle 9x-12 = -4x+8\)

Possible Answers:

\(\displaystyle \frac{4}{13}\)

\(\displaystyle -\frac{4}{13}\)

\(\displaystyle \frac{5}{13}\)

\(\displaystyle \frac{18}{13}\)

\(\displaystyle \frac{20}{13}\)

Correct answer:

\(\displaystyle \frac{20}{13}\)

Explanation:

Add 12 on both sides.

\(\displaystyle 9x-12 +12= -4x+8+12\)

\(\displaystyle 9x= -4x+20\)

Add \(\displaystyle 4x\) on both sides of the equation.

\(\displaystyle 9x+4x= -4x+20+4x\)

Simplify both sides.

\(\displaystyle 13x = 20\)

Divide by 13 on both sides.

\(\displaystyle \frac{13x }{13}= \frac{20}{13}\)

The answer is:  \(\displaystyle \frac{20}{13}\)

Example Question #2601 : Algebra Ii

Solve the equation:  \(\displaystyle 8x-9 = 31-8x\)

Possible Answers:

\(\displaystyle -\frac{5}{2}\)

\(\displaystyle \textup{No solution.}\)

\(\displaystyle \frac{5}{2}\)

\(\displaystyle \frac{11}{8}\)

\(\displaystyle -\frac{11}{8}\)

Correct answer:

\(\displaystyle \frac{5}{2}\)

Explanation:

Add \(\displaystyle 8x\) on both sides.

\(\displaystyle 8x-9 +8x= 31-8x+8x\)

The equation becomes:

\(\displaystyle 16x-9 = 31\)

Add nine on both sides.

\(\displaystyle 16x-9+9 = 31+9\)

The equation becomes:  

\(\displaystyle 16x = 40\)

Divide by 16 on both sides.

\(\displaystyle \frac{16x }{16}= \frac{40}{16}\)

Reduce the fractions.

\(\displaystyle \frac{40}{16} = \frac{8\times 5}{8\times 2}\)

The answer is:  \(\displaystyle \frac{5}{2}\)

Example Question #761 : Basic Single Variable Algebra

Solve the equation:  \(\displaystyle \frac{1}{3}x-\frac{1}{4}=\frac{3}{7}\)

Possible Answers:

\(\displaystyle \frac{47}{28}\)

\(\displaystyle -\frac{47}{28}\)

\(\displaystyle \frac{31}{28}\)

\(\displaystyle \frac{57}{28}\)

\(\displaystyle -\frac{15}{28}\)

Correct answer:

\(\displaystyle \frac{57}{28}\)

Explanation:

Determine the least common denominator of the fractions by multiplying all the denominators together.

\(\displaystyle 3(4)(7)= 84\)

We can multiply both sides by 84 in order to eliminate the fractions.

\(\displaystyle 84(\frac{1}{3}x-\frac{1}{4})=\frac{3}{7}\cdot 84\)

Simplify both sides.

\(\displaystyle 28x -21 = 3\cdot 12\)

\(\displaystyle 28x -21 = 36\)

Add 21 on both sides.

\(\displaystyle 28x = 57\)

Divide by 28 on both sides.

\(\displaystyle \frac{28x }{28}= \frac{57}{28}\)

The answer is:  \(\displaystyle \frac{57}{28}\)

Example Question #761 : Basic Single Variable Algebra

If 

\(\displaystyle f(x)=\frac{1}{3}x+4\) 

and 

\(\displaystyle g(x)=3x+4\),

what is the value of \(\displaystyle g(f(6))\)?

Possible Answers:

\(\displaystyle 12\)

\(\displaystyle 25\)

\(\displaystyle 22\)

\(\displaystyle 18\)

\(\displaystyle 15\)

Correct answer:

\(\displaystyle 22\)

Explanation:

In order to answer this question, you must first find \(\displaystyle f(6)\).

Plugging in 6 for x in the f(x) equation,

\(\displaystyle \\f(x)=\frac{1}{3}x+4 \\ \\f(6)=\frac{1}{3}(6)+4 \\ \\f(6)=\frac{2\cdot 3}{3}+4 \\ \\f(6)=2+4 \\ \\f(6)=6\)

you get 6 as your answer, so then you plug 6 in as the x in the g(x) equation.

\(\displaystyle \\g(x)=3x+4 \\g(6)=3(6)+4 \\g(6)=18+4 \\g(6)=22\).

Thus \(\displaystyle g(f(6))=22\).

Example Question #412 : Equations

Solve \(\displaystyle 4x-2x=4\) for \(\displaystyle x\).

Possible Answers:

\(\displaystyle x=2\)

\(\displaystyle x=0\)

\(\displaystyle x= -2\)

\(\displaystyle x=3\)

\(\displaystyle x=1\)

Correct answer:

\(\displaystyle x=2\)

Explanation:

\(\displaystyle 4x-2x=2x\), then setting \(\displaystyle 2x=4\) and dividing the entire equation by \(\displaystyle 2\), you get \(\displaystyle x=2\)

Example Question #413 : Equations

\(\displaystyle 4x+2x=?\)

Possible Answers:

\(\displaystyle 2\)

\(\displaystyle 6x\)

\(\displaystyle 2x\)

\(\displaystyle 6\)

\(\displaystyle 6x^{2}\)

Correct answer:

\(\displaystyle 6x\)

Explanation:

When adding \(\displaystyle 4x+2x\), you only add the coefficients and keep the variable, so\(\displaystyle 4+2=6\) and then keep the \(\displaystyle x\), so it equals \(\displaystyle 6x\).

Example Question #281 : Solving Equations

Solve the equation:   \(\displaystyle -8x = 9x+4\)

Possible Answers:

\(\displaystyle -4\)

\(\displaystyle \frac{4}{17}\)

\(\displaystyle \frac{13}{17}\)

\(\displaystyle -\frac{4}{17}\)

\(\displaystyle 4\)

Correct answer:

\(\displaystyle -\frac{4}{17}\)

Explanation:

Isolate the x-variable by subtracting \(\displaystyle 9x\) on both sides.

\(\displaystyle -8x-9x = 9x+4-9x\)

Simplify both sides.

\(\displaystyle -17x = 4\)

Divide by negative 17 on both sides.

\(\displaystyle \frac{-17x}{-17} =\frac{4 }{-17}\)

The answer is:  \(\displaystyle -\frac{4}{17}\)

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