How to find the solution to a rational equation with LCD - SAT Math

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In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

$\frac{1}{m^3}-\frac{1}{k^2}=\frac{1}{p}$

Answer

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Question

Solve for x:

$\frac{x+3}{7}=3$

Answer

The first step is to cancel out the denominator by multiplying both sides by 7:

$7\cdot \frac{x+3}{7}=3\cdot7$

Subtract 3 from both sides to get by itself:

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Question

Solve for and using elimination:

Answer

When using elimination, you need two factors to cancel out when the two equations are added together. We can get the in the first equation to cancel out with the in the second equation by multiplying everything in the second equation by :

Now our two equations look like this:

The can cancel with the , giving us:

These equations, when summed, give us:

Once we know the value for , we can just plug it into one of our original equations to solve for the value of :

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Question

Give the solution set of the rational equation $\frac{(x- 5)^{2}}{(x- 4) ^{2}} = 1$

Answer

Multiply both sides of the equation by the denominator $(x- 4) ^{2}$ :

$\frac{(x- 5)^{2}}{(x- 4) ^{2}} \cdot (x- 4) ^{2} =1 \cdot (x- 4) ^{2}$

$(x- 5)^{2} = (x- 4) ^{2}$

Rewrite both expression using the binomial square pattern:

$x^{2} - 2 \cdot x \cdot 5+ 5^{2} = x^{2} - 2 \cdot x \cdot 4+ 4^{2}$

$x^{2} -10x + 25 = x^{2} - 8x+16$

This can be rewritten as a linear equation by subtracting $x^{2}$ from both sides:

$x^{2} -10x + 25- x^{2} = x^{2} - 8x+16 - x^{2}$

Solve as a linear equation:

$\frac{-2x}{-2 } =\frac{ -9 }{-2}$

$x= \frac{9}{2}$

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Question

Solve:

$\frac{10-x}{40}=-5$

Answer

$\frac{10-x}{40}=-5$

Multiply by on each side

Subtract on each side

Multiply by on each side

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Question

In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

$\frac{1}{m^3}-\frac{1}{k^2}=\frac{1}{p}$

Answer

Compare your answer with the correct one above

Question

Solve for x:

$\frac{x+3}{7}=3$

Answer

The first step is to cancel out the denominator by multiplying both sides by 7:

$7\cdot \frac{x+3}{7}=3\cdot7$

Subtract 3 from both sides to get by itself:

Compare your answer with the correct one above

Question

Solve for and using elimination:

Answer

When using elimination, you need two factors to cancel out when the two equations are added together. We can get the in the first equation to cancel out with the in the second equation by multiplying everything in the second equation by :

Now our two equations look like this:

The can cancel with the , giving us:

These equations, when summed, give us:

Once we know the value for , we can just plug it into one of our original equations to solve for the value of :

Compare your answer with the correct one above

Question

Give the solution set of the rational equation $\frac{(x- 5)^{2}}{(x- 4) ^{2}} = 1$

Answer

Multiply both sides of the equation by the denominator $(x- 4) ^{2}$ :

$\frac{(x- 5)^{2}}{(x- 4) ^{2}} \cdot (x- 4) ^{2} =1 \cdot (x- 4) ^{2}$

$(x- 5)^{2} = (x- 4) ^{2}$

Rewrite both expression using the binomial square pattern:

$x^{2} - 2 \cdot x \cdot 5+ 5^{2} = x^{2} - 2 \cdot x \cdot 4+ 4^{2}$

$x^{2} -10x + 25 = x^{2} - 8x+16$

This can be rewritten as a linear equation by subtracting $x^{2}$ from both sides:

$x^{2} -10x + 25- x^{2} = x^{2} - 8x+16 - x^{2}$

Solve as a linear equation:

$\frac{-2x}{-2 } =\frac{ -9 }{-2}$

$x= \frac{9}{2}$

Compare your answer with the correct one above

Question

Solve:

$\frac{10-x}{40}=-5$

Answer

$\frac{10-x}{40}=-5$

Multiply by on each side

Subtract on each side

Multiply by on each side

Compare your answer with the correct one above

Question

Answer

Compare your answer with the correct one above

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Answer

Compare your answer with the correct one above

Question

In the equation below, , , and are non-zero numbers. What is the value of in terms of and ?

$\frac{1}{m^3}-\frac{1}{k^2}=\frac{1}{p}$

Answer

Compare your answer with the correct one above

Question

Solve for x:

$\frac{x+3}{7}=3$

Answer

The first step is to cancel out the denominator by multiplying both sides by 7:

$7\cdot \frac{x+3}{7}=3\cdot7$

Subtract 3 from both sides to get by itself:

Compare your answer with the correct one above

Question

Solve for and using elimination:

Answer

When using elimination, you need two factors to cancel out when the two equations are added together. We can get the in the first equation to cancel out with the in the second equation by multiplying everything in the second equation by :

Now our two equations look like this:

The can cancel with the , giving us:

These equations, when summed, give us:

Once we know the value for , we can just plug it into one of our original equations to solve for the value of :

Compare your answer with the correct one above

Question

Give the solution set of the rational equation $\frac{(x- 5)^{2}}{(x- 4) ^{2}} = 1$

Answer

Multiply both sides of the equation by the denominator $(x- 4) ^{2}$ :

$\frac{(x- 5)^{2}}{(x- 4) ^{2}} \cdot (x- 4) ^{2} =1 \cdot (x- 4) ^{2}$

$(x- 5)^{2} = (x- 4) ^{2}$

Rewrite both expression using the binomial square pattern:

$x^{2} - 2 \cdot x \cdot 5+ 5^{2} = x^{2} - 2 \cdot x \cdot 4+ 4^{2}$

$x^{2} -10x + 25 = x^{2} - 8x+16$

This can be rewritten as a linear equation by subtracting $x^{2}$ from both sides:

$x^{2} -10x + 25- x^{2} = x^{2} - 8x+16 - x^{2}$

Solve as a linear equation:

$\frac{-2x}{-2 } =\frac{ -9 }{-2}$

$x= \frac{9}{2}$

Compare your answer with the correct one above

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