Solved on Mar 10, 2024

Solve the equation $\frac{7}{x} + \frac{1}{4} = \frac{5}{x} + \frac{1}{3}$ and check the solution.

STEP 1

Assumptions
1. We are given the equation $\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}$ .
2. We need to find the value of $x$ that satisfies the equation.
3. We will check the solution by substituting the value of $x$ back into the original equation.

STEP 2

To solve the equation, we first need to find a common denominator for the fractions on both sides of the equation. The denominators are $x$ , 4, and 3.

STEP 3

Calculate the least common multiple (LCM) of the denominators $x$ , 4, and 3 to use as the common denominator. Since $x$ is an unknown, the LCM will be $12x$ .

STEP 4

Rewrite each fraction in the equation with the common denominator $12x$ .
$\frac{7}{x} \cdot \frac{12}{12} + \frac{1}{4} \cdot \frac{3x}{3x} = \frac{5}{x} \cdot \frac{12}{12} + \frac{1}{3} \cdot \frac{4x}{4x}$

STEP 5

Multiply each fraction by the appropriate form of 1 to get the common denominator.
$\frac{84}{12x} + \frac{3x}{12x} = \frac{60}{12x} + \frac{4x}{12x}$

STEP 6

Now that we have a common denominator, we can combine the fractions on each side of the equation.
$\frac{84 + 3x}{12x} = \frac{60 + 4x}{12x}$

STEP 7

Since the denominators are the same, we can set the numerators equal to each other.
$84 + 3x = 60 + 4x$

STEP 8

Subtract $3x$ from both sides to get the $x$ 's on one side.
$84 = 60 + x$

STEP 9

Subtract 60 from both sides to isolate $x$ .
$24 = x$

STEP 10

We have found the value of $x$ , which is 24. Now we need to check the solution by substituting $x = 24$ back into the original equation.
$\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}$

STEP 11

Substitute $x = 24$ into the left side of the original equation.
$\frac{7}{24}+\frac{1}{4}$

STEP 12

Find a common denominator for the fractions on the left side, which is 24.
$\frac{7}{24}+\frac{6}{24}$

STEP 13

Combine the fractions on the left side.
$\frac{7 + 6}{24} = \frac{13}{24}$

STEP 14

Substitute $x = 24$ into the right side of the original equation.
$\frac{5}{24}+\frac{1}{3}$

STEP 15

Find a common denominator for the fractions on the right side, which is 24.
$\frac{5}{24}+\frac{8}{24}$

STEP 16

Combine the fractions on the right side.
$\frac{5 + 8}{24} = \frac{13}{24}$

STEP 17

Since both the left side and the right side of the original equation are equal when $x = 24$ , the solution checks out.
The solution to the equation is $x = 24$ .

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Solve the equation $\frac{7}{x} + \frac{1}{4} = \frac{5}{x} + \frac{1}{3}$ and check the solution.

STEP 1

Assumptions
1. We are given the equation $\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}$ .
2. We need to find the value of $x$ that satisfies the equation.
3. We will check the solution by substituting the value of $x$ back into the original equation.

STEP 2

To solve the equation, we first need to find a common denominator for the fractions on both sides of the equation. The denominators are $x$ , 4, and 3.

STEP 3

Calculate the least common multiple (LCM) of the denominators $x$ , 4, and 3 to use as the common denominator. Since $x$ is an unknown, the LCM will be $12x$ .

STEP 4

Rewrite each fraction in the equation with the common denominator $12x$ .
$\frac{7}{x} \cdot \frac{12}{12} + \frac{1}{4} \cdot \frac{3x}{3x} = \frac{5}{x} \cdot \frac{12}{12} + \frac{1}{3} \cdot \frac{4x}{4x}$

STEP 5

Multiply each fraction by the appropriate form of 1 to get the common denominator.
$\frac{84}{12x} + \frac{3x}{12x} = \frac{60}{12x} + \frac{4x}{12x}$

STEP 6

Now that we have a common denominator, we can combine the fractions on each side of the equation.
$\frac{84 + 3x}{12x} = \frac{60 + 4x}{12x}$

STEP 7

Since the denominators are the same, we can set the numerators equal to each other.
$84 + 3x = 60 + 4x$

STEP 8

Subtract $3x$ from both sides to get the $x$ 's on one side.
$84 = 60 + x$

STEP 9

Subtract 60 from both sides to isolate $x$ .
$24 = x$

STEP 10

We have found the value of $x$ , which is 24. Now we need to check the solution by substituting $x = 24$ back into the original equation.
$\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}$

STEP 11

Substitute $x = 24$ into the left side of the original equation.
$\frac{7}{24}+\frac{1}{4}$

STEP 12

Find a common denominator for the fractions on the left side, which is 24.
$\frac{7}{24}+\frac{6}{24}$

STEP 13

Combine the fractions on the left side.
$\frac{7 + 6}{24} = \frac{13}{24}$

STEP 14

Substitute $x = 24$ into the right side of the original equation.
$\frac{5}{24}+\frac{1}{3}$

STEP 15

Find a common denominator for the fractions on the right side, which is 24.
$\frac{5}{24}+\frac{8}{24}$

STEP 16

Combine the fractions on the right side.
$\frac{5 + 8}{24} = \frac{13}{24}$

STEP 17

Since both the left side and the right side of the original equation are equal when $x = 24$ , the solution checks out.
The solution to the equation is $x = 24$ .

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Solve the equation 7x+14=5x+13\frac{7}{x} + \frac{1}{4} = \frac{5}{x} + \frac{1}{3}x7​+41​=x5​+31​ and check the solution.

STEP 1

Assumptions1. We are given the equation 7x+14=5x+13\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}x7​+41​=x5​+31​.2. We need to find the value of xxx that satisfies the equation.3. We will check the solution by substituting the value of xxx back into the original equation.

STEP 2

To solve the equation, we first need to find a common denominator for the fractions on both sides of the equation. The denominators are xxx, 4, and 3.

STEP 3

Calculate the least common multiple (LCM) of the denominators xxx, 4, and 3 to use as the common denominator. Since xxx is an unknown, the LCM will be 12x12x12x.

STEP 4

STEP 5

Multiply each fraction by the appropriate form of 1 to get the common denominator.8412x+3x12x=6012x+4x12x \frac{84}{12x} + \frac{3x}{12x} = \frac{60}{12x} + \frac{4x}{12x} 12x84​+12x3x​=12x60​+12x4x​

STEP 6

Now that we have a common denominator, we can combine the fractions on each side of the equation.84+3x12x=60+4x12x \frac{84 + 3x}{12x} = \frac{60 + 4x}{12x} 12x84+3x​=12x60+4x​

STEP 7

Since the denominators are the same, we can set the numerators equal to each other.84+3x=60+4x 84 + 3x = 60 + 4x 84+3x=60+4x

STEP 8

Subtract 3x3x3x from both sides to get the xxx's on one side.84=60+x 84 = 60 + x 84=60+x

STEP 9

Subtract 60 from both sides to isolate xxx.24=x 24 = x 24=x

STEP 10

We have found the value of xxx, which is 24. Now we need to check the solution by substituting x=24x = 24x=24 back into the original equation.7x+14=5x+13 \frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3} x7​+41​=x5​+31​

STEP 11

Substitute x=24x = 24x=24 into the left side of the original equation.724+14 \frac{7}{24}+\frac{1}{4} 247​+41​

STEP 12

Find a common denominator for the fractions on the left side, which is 24.724+624 \frac{7}{24}+\frac{6}{24} 247​+246​

STEP 13

Combine the fractions on the left side.7+624=1324 \frac{7 + 6}{24} = \frac{13}{24} 247+6​=2413​

STEP 14

Substitute x=24x = 24x=24 into the right side of the original equation.524+13 \frac{5}{24}+\frac{1}{3} 245​+31​

STEP 15

Find a common denominator for the fractions on the right side, which is 24.524+824 \frac{5}{24}+\frac{8}{24} 245​+248​

STEP 16

Combine the fractions on the right side.5+824=1324 \frac{5 + 8}{24} = \frac{13}{24} 245+8​=2413​

STEP 17

Since both the left side and the right side of the original equation are equal when x=24x = 24x=24, the solution checks out.The solution to the equation is x=24x = 24x=24.

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Solve the equation $\frac{7}{x} + \frac{1}{4} = \frac{5}{x} + \frac{1}{3}$ and check the solution.

Assumptions
1. We are given the equation $\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}$ .
2. We need to find the value of $x$ that satisfies the equation.
3. We will check the solution by substituting the value of $x$ back into the original equation.

To solve the equation, we first need to find a common denominator for the fractions on both sides of the equation. The denominators are $x$ , 4, and 3.

Calculate the least common multiple (LCM) of the denominators $x$ , 4, and 3 to use as the common denominator. Since $x$ is an unknown, the LCM will be $12x$ .

Multiply each fraction by the appropriate form of 1 to get the common denominator.
$\frac{84}{12x} + \frac{3x}{12x} = \frac{60}{12x} + \frac{4x}{12x}$

Now that we have a common denominator, we can combine the fractions on each side of the equation.
$\frac{84 + 3x}{12x} = \frac{60 + 4x}{12x}$

Since the denominators are the same, we can set the numerators equal to each other.
$84 + 3x = 60 + 4x$

Subtract $3x$ from both sides to get the $x$ 's on one side.
$84 = 60 + x$

Subtract 60 from both sides to isolate $x$ .
$24 = x$

We have found the value of $x$ , which is 24. Now we need to check the solution by substituting $x = 24$ back into the original equation.
$\frac{7}{x}+\frac{1}{4}=\frac{5}{x}+\frac{1}{3}$

Substitute $x = 24$ into the left side of the original equation.
$\frac{7}{24}+\frac{1}{4}$

Find a common denominator for the fractions on the left side, which is 24.
$\frac{7}{24}+\frac{6}{24}$

Combine the fractions on the left side.
$\frac{7 + 6}{24} = \frac{13}{24}$

Substitute $x = 24$ into the right side of the original equation.
$\frac{5}{24}+\frac{1}{3}$

Find a common denominator for the fractions on the right side, which is 24.
$\frac{5}{24}+\frac{8}{24}$

Combine the fractions on the right side.
$\frac{5 + 8}{24} = \frac{13}{24}$

Since both the left side and the right side of the original equation are equal when $x = 24$ , the solution checks out.
The solution to the equation is $x = 24$ .