Problem 101 - Intermediate Algebra

Question

Solve each equation. $$ \frac{4}{5}(x+5)=\frac{7}{8}(3 x+23)-7 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = -5 \).

Step 1: Eliminate fractions by finding a common denominator

Both sides of the equation have fractions. The denominators are 5 and 8, so the least common denominator (LCD) is 40. Multiply every term by 40 to eliminate the fractions: \[ 40 \times \frac{4}{5}(x+5) = 40 \times \frac{7}{8}(3x+23) - 40 \times 7 \] This simplifies to \[ 32(x+5) = 35(3x+23) - 280 \].

Step 2: Distribute and simplify each side

Distribute the constants into each bracket:On the left:\[ 32(x+5) = 32x + 160 \]On the right:\[ 35(3x + 23) = 105x + 805 \].So our equation is now:\[ 32x + 160 = 105x + 805 - 280 \].

Step 3: Simplify and collect like terms

First, simplify the right side by combining like terms:\[ 105x + 805 - 280 = 105x + 525 \].Now the equation looks like:\[ 32x + 160 = 105x + 525 \].

Step 4: Isolate the variable term

To isolate the term with \( x \) on one side, subtract \( 32x \) from both sides of the equation:\[ 160 = 105x - 32x + 525 \],which simplifies to:\[ 160 = 73x + 525 \].

Step 5: Solve for the variable

Subtract 525 from both sides to isolate the \( x \) term:\[ 160 - 525 = 73x \],which simplifies to:\[ -365 = 73x \].Finally, divide both sides by 73:\[ x = \frac{-365}{73} \],which simplifies to:\[ x = -5 \].

Key Concepts

Eliminating Fractions

Fractions in equations can often seem daunting, but eliminating them simplifies the solving process. To remove fractions from an equation, we use a method called clearing fractions.

• First, identify the denominators of the fractions. In our original equation, the denominators are 5 and 8.
• Next, find a least common denominator (LCD), which in this case is 40. This means 40 is the smallest number that both denominators can divide into evenly.
• Multiply each term in the equation by this LCD. By doing this, you essentially convert the fractions into whole numbers, making calculations much more straightforward.

After multiplying through by 40, the equation no longer contains any fractions, simplifying it to: \[ 32(x + 5) = 35(3x + 23) - 280 \]. This method drastically reduces the complexity of the problem.

Distributive Property

The distributive property is a fundamental algebraic principle used to expand expressions by distributing a multiplying factor across terms inside parentheses.

• Apply the distributive property by multiplying the factor outside the parentheses by each term inside the parentheses.
• For example, applying this property to the left side of \( 32(x + 5) \) gives \( 32x + 160 \).

On the right side of our equation, we have \( 35(3x + 23) \), which simplifies to \( 105x + 805 \). This step is critical as it converts all terms into a linear form, allowing us to handle like terms more easily in the next steps of solving the equation.

Combining Like Terms

Combining like terms is an essential skill in simplifying equations. Like terms have identical variable components.

• Look at both sides of the equation to spot like terms. For instance, \( 105x \) and \( 32x \) are like terms since they both include the variable \( x \).
• On the right side of our equation, we originally had \( 105x + 805 - 280 \). Combine terms without variables first: \( 805 - 280 = 525 \), resulting in the simplified equation \( 105x + 525 \).

This step reduces the clutter in the equation, focusing on the key components necessary for isolating the variable.

Isolating Variable

The main goal of solving any equation is to isolate the variable, typically on one side of the equation, leading to a simple expression like \( x = c \), where \( c \) is a constant.

• In the equation \( 32x + 160 = 105x + 525 \), start by moving the term with \( x \) to one side. Subtract \( 32x \) from both sides, giving \( 160 = 73x + 525 \).
• Next, isolate \( 73x \) by subtracting 525 from both sides: \( 160 - 525 = 73x \), which simplifies to \( -365 = 73x \).

Finally, divide both sides by 73 to solve for \( x \): \( x = \frac{-365}{73} \), leading to \( x = -5 \). This method of isolation allows for a clear pathway to the solution, breaking down the equation step-by-step until the variable is alone.