Problem 66 - Algebra for College Students

Question

Solve each equation, and check the solution. \(\frac{3 x+2}{7}-\frac{x+4}{5}=2\)

Step-by-Step Solution

Answer

x = 11

Step 1 - Eliminate the fractions

Multiply each term by the least common denominator (LCD) of the fractions. The LCD of 7 and 5 is 35. \[\frac{3x + 2}{7} \times 35 - \frac{x + 4}{5} \times 35 = 2 \times 35\]This simplifies to \[5(3x + 2) - 7(x + 4) = 70\]

Step 2 - Distribute and simplify

Distribute the 5 and the 7 in the equation: \[15x + 10 - 7x - 28 = 70\]Combine like terms: \[8x - 18 = 70\]

Step 3 - Solve for x

Add 18 to both sides of the equation to isolate the term with x: \[8x - 18 + 18 = 70 + 18\]This simplifies to \[8x = 88\]Divide by 8: \[x = 11\]

Step 4 - Check the solution

Substitute \(x = 11\) back into the original equation and check if both sides are equal. The original equation is: \[\frac{3x + 2}{7} - \frac{x + 4}{5} = 2\]Substitute \(x = 11\): \[\frac{3(11) + 2}{7} - \frac{11 + 4}{5} = 2\]Simplify: \[\frac{33 + 2}{7} - \frac{15}{5} = 2\]\[\frac{35}{7} - 3 = 2\]\[5 - 3 = 2\]\[2 = 2\]Since both sides are equal, \(x = 11\) is the correct solution.

Key Concepts

The Least Common Denominator (LCD)

When solving rational equations, one of the first steps is to eliminate the fractions. This makes the equation easier to manage. To do this, we multiply every term by the Least Common Denominator (LCD) of all the fractions involved.
The LCD is the smallest number that each of the denominators can divide into without a remainder.
For the equation \(\frac{3 x+2}{7}-\frac{x+4}{5}=2\), the denominators are 7 and 5.
The smallest number that both 7 and 5 divide into evenly is 35. So, 35 is the LCD.
Once we have the LCD, we multiply every term in the equation by 35. This step helps to cancel out the denominators:
\(\frac{3x + 2}{7} \times 35 - \frac{x + 4}{5} \times 35 = 2 \times 35\)
Simplifying each of these terms, we get:
\[5(3x + 2) - 7(x + 4) = 70\] Now we have an equation without fractions, which makes it easier to solve.

The Distributive Property

After eliminating the fractions in a rational equation, the next step is to simplify the equation.
This often involves using the distributive property. The distributive property helps us to remove parentheses by multiplying each term inside the parentheses by the number outside.
For instance, in the equation: \[5(3x + 2) - 7(x + 4) = 70\] We distribute 5 and -7 to each term inside their respective parentheses:
\[15x + 10 - 7x - 28 = 70\]
Now, the equation is easier to understand and solve, as it becomes:
\[15x + 10 - 7x - 28 = 70\] By combining like terms, we get: \[8x - 18 = 70\]

Isolating Variables

The final step in solving the rational equation is isolating the variable.
This means getting the variable x by itself on one side of the equation.
From our simplified equation: \[8x - 18 = 70\] To isolate x, we first add 18 to both sides to move the constant term away:
\[8x - 18 + 18 = 70 + 18\] Simplifying, we get: \[8x = 88\] Next, we need to solve for x by dividing both sides of the equation by 8: \[x = 11\] Now that we have isolated x and found its value, we need to check our solution by substituting it back into the original equation: \[\frac{3x + 2}{7} - \frac{x + 4}{5} = 2\] When we plug in 11 for x and simplify both sides, we find that our solution is correct: \[2 = 2\]