GMAT Math : Equations

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Example Question #51 : Solving Equations

Solve for in the equation

$a^{3}- y + 1= c^{2}$

Possible Answers:

$a=\sqrt[3]{ c^{2}} + y - 1$ or $a=-\sqrt[3]{ c^{2}} + y - 1$

$a=\sqrt[3]{ c^{2}} + y - 1$ or $a=-\sqrt[3]{ c^{2}} - y + 1$

$a=\sqrt[3]{ c^{2}} + y - 1$

$a=\sqrt[3]{ c^{2} + y - 1}$ or $a= -\sqrt[3]{ c^{2} + y - 1}$

$a=\sqrt[3]{ c^{2} + y - 1}$

Correct answer:

$a=\sqrt[3]{ c^{2} + y - 1}$

Explanation:

$a^{3}- y + 1= c^{2}$

$a^{3}- y + 1+ y - 1= c^{2} + y - 1$

$a^{3} = c^{2} + y - 1$

$\sqrt[3]{ a^{3} } =\sqrt[3]{ c^{2} + y - 1}$

$a=\sqrt[3]{ c^{2} + y - 1}$

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Example Question #52 : Solving Equations

Solve for in the equation:

$v^{2}+ 4vt + 4t^{2} = h+30$

Possible Answers:

$t=\frac{- v + h + 60 }{2}$ or $t=\frac{- v - h - 60 }{2}$

$t=\frac{- v + h + 30 }{2}$ or $t=\frac{- v - h - 30 }{2}$

t=- 2v + h + 30 or t=- 2v - h - 30

$t=\frac{- v + \sqrt{ h + 30 }}{2}$ or $t=\frac{- v - \sqrt{ h + 30 } }{2}$ .

$t=- 2v + \sqrt{ h + 30 }$ or $t=- 2v - \sqrt{ h + 30 }$

Correct answer:

$t=\frac{- v + \sqrt{ h + 30 }}{2}$ or $t=\frac{- v - \sqrt{ h + 30 } }{2}$ .

Explanation:

$v^{2}+ 4vt + 4t^{2}$ is a perfect square trinomial:

$v^{2}+ 4vt + 4t^{2} = v^{2}+ 2 \cdot 2 t + (2t)^{2} = (v+2t)^{2}$

The equation can be rewritten as

$(v+2t)^{2} = h + 30$

By the square-root property, since no assumption was made about the sign of any variable,

$v+2t = \pm\sqrt{ h + 30 }$

$v+2t -v = \pm \sqrt{ h + 30 }-v$

$2t= -v\pm \sqrt{ h + 30 }$

$\frac{2t}{2}=\frac{- v\pm \sqrt{ h + 30 } }{2}$

$t=\frac{- v\pm \sqrt{ h + 30 }}{2}$

Therefore,

$t=\frac{- v + \sqrt{ h + 30 }}{2}$ or $t=\frac{- v - \sqrt{ h + 30 } }{2}$ .

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Example Question #51 : Solving Equations

Solve for in the equation

$y^{2} + 6y + Q = 0$

Possible Answers:

$y = -3 - \sqrt{9-Q}$ or $y = -3 \p+\sqrt{9-Q}$

$y = 3 + \sqrt{9+Q}$ or $y = -3 +\sqrt{9+Q}$

$y = -3 - \sqrt{9+Q}$ or $y = -3 \p+\sqrt{9+Q}$

$y = 3 - \sqrt{9-Q}$ or $y = -3 -\sqrt{9-Q}$

$y = 3 - \sqrt{9-Q}$ or $y = 3 \p+\sqrt{9-Q}$

Correct answer:

$y = -3 - \sqrt{9-Q}$ or $y = -3 \p+\sqrt{9-Q}$

Explanation:

The statement is a quadratic equation in , so it can be solved using the quadratic formula,

$y = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

where a = 1,b = 6, c = Q

$y = \frac{-6 \pm \sqrt{6^{2}-4(1)(Q)}}{2 (1)}$

$y = \frac{-6 \pm \sqrt{36-4Q}}{2}$

$y = \frac{-6 \pm \sqrt{4(9-Q)}}{2}$

$y = \frac{-6 \pm 2 \sqrt{9-Q}}{2}$

$y = \frac{2\left (-3 \pm \sqrt{9-Q} \right )}{2}$

$y = -3 \pm \sqrt{9-Q}$

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Example Question #54 : Solving Equations

How many distinct solutions are there to the following equation?

$\small 3x^2-27=0$

Possible Answers:

Infinitely Many

Correct answer:

Explanation:

We are given a classic quadratic equation, but we aren't asked for the solutions, just how many distinct solutions there are. Remember, distinct solutions are different solutions. If we get two solutions that are the same numbers, they do not count.

The quickest way to solve this involves some factoring.

Start by pulling out a 3

$\small 3(x^2-9)=0$

Now, within our parentheses, we have a classic difference of squares. The interior factors further to look like this.

$\small \small 3{(x+3)(x-3)}=0$

From here we can either solve the equation and count our solutions, or we can recognize that the two factors are different and therefore will give different solutions. Let's solve it by using the Zero Product Property

Solution 1

$\small x+3=0$

$\small x=-3$

Solution 2

$\small x-3=0$

$\small x=3$

Thus, we have two distinct solutions!

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Example Question #55 : Solving Equations

Solve for in the equation

$h = f+ \sqrt{3g+ 4}$

Possible Answers:

$y =\frac{ \sqrt{h - f} - 4}{3}$

$g= \frac{h^{2}-2hf+f^{2} - 12}{3}$

$g= \frac{h^{2}-2hf+f^{2} - 4}{3}$

$g= \frac{h^{2} +f^{2} - 4}{3}$

$y =\frac{ \sqrt{h - f} -12}{3}$

Correct answer:

$g= \frac{h^{2}-2hf+f^{2} - 4}{3}$

Explanation:

$h = f+ \sqrt{3g+ 4}$

$h- f = f+ \sqrt{3g+ 4} - f$

$\sqrt{3g+ 4} = h- f$

$\left (\sqrt{3g+ 4} \right )^{2}= \left (h- f \right )^{2}$

$3g+4= h^{2}-2hf+f^{2}$

$3g+4- 4= h^{2}-2hf+f^{2} - 4$

$3g = h^{2}-2hf+f^{2} - 4$

$\frac{3g }{3}= \frac{h^{2}-2hf+f^{2} - 4}{3}$

$g= \frac{h^{2}-2hf+f^{2} - 4}{3}$

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Example Question #56 : Solving Equations

is 44% of .

2K+3P is what percent of ?

Possible Answers:

$121 \frac{1}{3} \%$

$388 \%$

$135 \%$

$332\%$

$314\frac{2}{3} \%$

Correct answer:

$388 \%$

Explanation:

is 44% of , so is $2 \times 44 \% = 88 \%$ of .

Also, is 300% of .

Add these:

2K+3P is $88 \% + 300 \% = 388 \%$ of

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Example Question #57 : Solving Equations

Solve for :

-3x+18=7x-2

Possible Answers:

x=-4

x=2

x=-2

x=5

x=-5

Correct answer:

x=2

Explanation:

To solve the equation, we first group the terms on one side and the constants on the other side:

-3x+18=7x-2

-10x=-20

Now we can simply divide both sides by to solve for :

-10x=-20

x=2

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Example Question #371 : Algebra

Solve the following equation:

2x-5+3x=18-6x+10

Possible Answers:

x=6

x=3

x=11

x=1

x=5

Correct answer:

x=3

Explanation:

To solve the equation, we must simplify it by adding together the like terms. We can group the terms on the left side of the equation and the constants on the right side of the equation:

2x-5+3x=18-6x+10

2x+3x+6x=18+10+5

11x=33

$x=\frac{33}{11}$

x=3

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Example Question #372 : Algebra

Solve the following equation for :

3x+5=26

Possible Answers:

$x=\frac{31}{3}$

x=6

x=7

x=26

x=0

Correct answer:

x=7

Explanation:

To solve, we must isolate . First, subtract from both sides and then divide both sides by .

3x+5-5=26-5

3x=21

$\frac{3x}{3}=\frac{21}{3}$

x=7

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Example Question #51 : Solving Equations

Solve for :

2x+5=23

Possible Answers:

x=16

x=2

x=14

x=9

Correct answer:

x=9

Explanation:

To solve, isolate .

2x+5=23

2x=18

x=9

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GMAT Math : Equations

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #51 : Solving Equations

Example Question #52 : Solving Equations

Example Question #51 : Solving Equations

Example Question #54 : Solving Equations

Example Question #55 : Solving Equations

Example Question #56 : Solving Equations

Example Question #57 : Solving Equations

Example Question #371 : Algebra

Example Question #372 : Algebra

Example Question #51 : Solving Equations

All GMAT Math Resources

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