Question

Solve the equation and check your solution. (Some equations have no solution.) $$ \frac{100-4 u}{3}=\frac{5 u+6}{4}+6 $$

Short answer

The solution to the equation is \( u = 10 \).

Step-by-step solution

Simplify the equation

Begin by getting rid of the fractions. To do this, one needs to find a common multiple of 3 and 4. 3 times 4 equals 12, which is their common multiple, so both sides of the equation should be multiplied by 12. This will give: \(12 \times \frac{100-4u}{3}= 12 \times (\frac{5u+6}{4}+6)\). After multiplying, the equation becomes: \(400-16u=3(5u+6)+72\).

Distribute and simplify

The next step is to distribute the 3 on the right side of the equation. This will give: \(400-16u=15u+18+72\). Then combine like terms to simplify, the equation becomes: \(400-16u=15u+90\).

Solve for the variable

Now, the aim is to isolate \( u \) to one side of the equation. Start doing this by adding \( 16u \) to both sides of the equation to get the expression: \(400=31u+90\). Then isolate \( u \) by taking \( 90 \) to other side of the equation by subtracting both sides by 90, this gives: \(310=31u\). Finally, solve for \( u \) by dividing both sides by 31 to get the value of \( u \). This gives: \(u=10\).

Check the solution

The final step is to verify the solution. Substitute \( u=10 \) back into the original equation and see if the both sides equal. This gives: \(\frac{100-4(10)}{3}=?\frac{5(10)+6}{4}+6\) which simplifies to \(\frac{60}{3}=?\frac{56}{4}+6\) then to \(20 =? 14+6\) finally to \(20 = 20\), which is true. Therefore, \(u=10\) is indeed the correct solution to the equation.

Key concepts

Fractions and Their Role in Equations

Fractions are an important component of algebra that often represent partial quantities. When solving equations with fractions, the goal is to simplify the equation for easier manipulation.
In the given problem, \[\frac{100-4u}{3}=\frac{5u+6}{4}+6\] fractions appear on both sides. The main challenge is eliminating these fractions, which is done by finding a common multiple of the denominators to multiply the entire equation. By doing this, we essentially convert the equation into one without fractions.
For denominators 3 and 4, the least common multiple is 12, which allows us to multiply every term by 12, like this:\[12 \times \left(\frac{100-4u}{3}\right)= 12 \times \left(\frac{5u+6}{4}+6\right)\]
This multiplication gets rid of the fractions, converting the equation to one involving whole numbers, making it easier to solve.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra used to simplify and solve equations. It involves distributing a multiplier over terms inside parentheses, making the expression easier to handle.
In the original problem, after removing the fractions, the right side of the equation shows:\[3(5u+6)+72\]
The distributive property allows us to multiply 3 by each term inside the parentheses:\[15u + 18 + 72\]
By distributing, we simplify the equation and can combine like terms. This transformation is crucial for the next steps in solving, which involves isolating the variable.

Importance of Checking Solutions

After solving an equation, it's essential to check if the solution is correct. This step ensures no mistakes were made during manipulation.
For the equation \[\frac{100-4u}{3}=\frac{5u+6}{4}+6\]with the solution found as \(u=10\), we substitute back into the original equation:- Substitute: \[\frac{100-4(10)}{3} = \frac{5(10)+6}{4} + 6\]- Simplifies to:\[\frac{60}{3} = \frac{56}{4} + 6\],- Which further simplifies to:\[20 = 14 + 6\] - And finally confirms: \[20 = 20\]
If both sides of the equation equal, it verifies the solution is correct. Checking solutions prevents errors and helps solidify understanding of the equation-solving process.