Problem 57 - Algebra 2

Question

Solve each equation. Check your solution. $$ \frac{2}{5} x+\frac{3}{7}=1-\frac{4}{7} x $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{10}{17}\).

Step 1: Eliminate Fractions

To simplify the equation, get rid of the fractions by finding a common denominator. The least common multiple of 5 and 7 is 35. Multiply every term by 35 to eliminate the fractions: \[ 35 \left( \frac{2}{5}x \right) + 35 \left( \frac{3}{7} \right) = 35 \left( 1 \right) - 35 \left( \frac{4}{7}x \right) \].This results in:\[ 14x + 15 = 35 - 20x \].

Step 2: Combine Like Terms

Move all terms involving \(x\) to one side of the equation and constant terms to the other side:\[ 14x + 20x = 35 - 15 \].This simplifies to:\[ 34x = 20 \].

Step 3: Solve for x

Now, solve for \(x\) by dividing both sides by 34:\[ x = \frac{20}{34} \].Simplify the fraction to get:\[ x = \frac{10}{17} \].

Step 4: Check the Solution

Substitute \(x = \frac{10}{17}\) back into the original equation to verify the solution:\[ \frac{2}{5} \times \frac{10}{17} + \frac{3}{7} = 1 - \frac{4}{7} \times \frac{10}{17} \].Calculate each term:- Left side: \(\frac{4}{17} + \frac{3}{7}\).- Right side: \(1 - \frac{40}{119}\).Check if both sides are equal:\[ \frac{4 \times 7 + 3 \times 17}{119} = \frac{119 - 40}{119} \].Simplify:\[ \frac{28 + 51}{119} = \frac{79}{119} \] and \[ \frac{79}{119} \].Both sides are equal, confirming \(x = \frac{10}{17}\) is correct.

Key Concepts

Fractions

Fractions are portions of a whole, represented as two integers: the numerator (top number) and the denominator (bottom number). In this equation, fractions are utilized to represent parts of variables and constants, such as \( \frac{2}{5}x \) and \( \frac{3}{7} \). When dealing with fractions in linear equations, it can be complex due to different denominators.
To begin solving equations with fractions, it's often easiest to eliminate them by finding a way to convert each term into whole numbers. This process typically involving multiplying each term in the equation by the least common denominator of all fractions involved. Doing so makes the equation simpler and easier to manage, as we will see in the next sections.

Common Denominator

To effectively handle fractions in equations, finding a common denominator is crucial. The common denominator allows you to add, subtract, or compare fractions by converting them into equivalent fractions with the same bottom number.
In this exercise, the fractions have denominators of 5 and 7. The least common multiple of these numbers is 35. Thus, 35 becomes our common denominator. By multiplying each term in the equation by 35, you can remove the fractions entirely:

• Multiply \( \frac{2}{5}x \) by 35, yielding 14x.
• Multiply \( \frac{3}{7} \) by 35, yielding 15.
• Multiply \( 1 \) by 35, yielding 35.
• Multiply \( -\frac{4}{7}x \) by 35, yielding -20x.

With these calculations, the equation becomes \( 14x + 15 = 35 - 20x \). This simplification paves the way for solving the equation without fractions, making the process less cumbersome.

Simplifying Equations

Simplifying an equation involves combining like terms and isolating the variable of interest. Once we have eliminated the fractions in the equation, it turns into something much more manageable: \( 14x + 15 = 35 - 20x \).
The first step in this process is to get all terms involving \(x\) on one side of the equation and constants on the other. We achieve this by:

• Adding \( 20x \) to both sides, which results in \( 14x + 20x = 35 - 15 \).

This operation simplifies to \( 34x = 20 \). The next step is dividing both sides by 34 to isolate \(x\). This results in \( x = \frac{20}{34} \).
Finally, simplify the fraction to \( x = \frac{10}{17} \). At this stage, we have successfully simplified and solved the equation.

Checking Solutions

After finding the solution to an equation, it’s important to verify its correctness by checking the solution. This ensures no errors were made during calculations.
To check the solution of this equation, substitute \( x = \frac{10}{17} \) back into the original equation:

• Calculate the left side: \( \frac{2}{5} \times \frac{10}{17} + \frac{3}{7} \), simplifying to \( \frac{4}{17} + \frac{3}{7} \).
• Calculate the right side: \( 1 - \frac{4}{7} \times \frac{10}{17} \), which simplifies to \( \frac{79}{119} \).

Both sides must equal for the solution to be valid.
By comparing:

• Combine fractions on the left: \( \frac{4 \times 7 + 3 \times 17}{119} = \frac{79}{119} \).

Both fractions match, confirming that \( x = \frac{10}{17} \) is indeed the correct solution.