Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$\frac{13+2 x}{4 x+1}=\frac{3}{4}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{49}{4} \).

Step 1: Cross Multiply

The given equation is \( \frac{13+2x}{4x+1} = \frac{3}{4} \). To eliminate the fractions, cross multiply to get: \( 4(13+2x) = 3(4x+1) \).

Step 2: Distribute

Distribute the numbers across the parentheses. On the left side, distribute 4: \( 4 \times 13 + 4 \times 2x = 52 + 8x \). On the right side, distribute 3: \( 3 \times 4x + 3 \times 1 = 12x + 3 \).

Step 3: Simplify both sides

Rewrite the equation after distribution: \( 52 + 8x = 12x + 3 \).

Step 4: Isolate variable terms

Subtract \( 8x \) from both sides to isolate x terms: \( 52 = 12x - 8x + 3 \) becomes \( 52 = 4x + 3 \).

Step 5: Solve for x

Subtract 3 from both sides to further isolate x: \( 52 - 3 = 4x \) becomes \( 49 = 4x \). Now, divide by 4: \( \frac{49}{4} = x \).

Step 6: Verify the solution

Substitute \( x = \frac{49}{4} \) back into the original equation to verify: The left side becomes \( \frac{13 + 2\left(\frac{49}{4}\right)}{4\left(\frac{49}{4}\right) + 1} = \frac{13 + \frac{98}{4}}{49 + 1} = \frac{39.5}{50} \), simplifying gives exactly \( \frac{3}{4} \), verifying the solution is correct.

Key Concepts

Cross multiplication

Cross multiplication is a handy technique that helps in solving equations involving fractions, particularly those set equal to each other, when the equation is in the form of \(\frac{a}{b} = \frac{c}{d}\). This method allows us to eliminate the fractions by multiplying diagonally. In essence, we multiply both sides of the equation by the denominators.
Using our original exercise as an example: \(\frac{13+2x}{4x+1} = \frac{3}{4}\), cross multiplying gives us the equation \(4(13+2x) = 3(4x+1)\).
Here are the quick steps to perform cross multiplication:

• Identify the numerators and denominators on both sides of the equation.
• Multiply the numerator of the first fraction by the denominator of the second fraction.
• Multiply the numerator of the second fraction by the denominator of the first fraction.
• Set these two products equal to each other.

This results in a simple equation without fractions, which is much easier to handle.

Distributive property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations. The property states that \(a(b + c) = ab + ac\). This means we can distribute the multiplication over addition or subtraction within parentheses.
In our given equation, after cross multiplying, we have: \(4(13+2x) = 3(4x+1)\).
We apply the distributive property as follows:

• Distribute 4 to both 13 and \(2x\), resulting in \(52 + 8x\).
• Distribute 3 to both \(4x\) and 1, resulting in \(12x + 3\).

This step is crucial as it simplifies the equation to a point where you can easily combine like terms and proceed towards solving the equation for the variable.

Solving linear equations

Solving linear equations involves isolating the variable to one side of the equation, making it the subject of the formula. After distributing, our equation simplifies to \(52 + 8x = 12x + 3\).
We proceed by isolating variable terms:

• Subtract \(8x\) from both sides to get all terms involving \(x\) on one side: \(52 = 4x + 3\).
• Subtract 3 from both sides to further isolate terms involving the variable: \(49 = 4x\).
• Finally, divide both sides by 4 to solve for \(x\): \(x = \frac{49}{4}\).

Remember, the key steps are to combine like terms, move variable terms to one side, and isolate the variable completely to solve the equation.

Verification of solution

Verification of solution is an important part of solving any equation, which ensures that your solution is correct. Once you have found the value of the variable, substitute it back into the original equation to check your work.
In our example, having solved for \(x\) yielding \(x = \frac{49}{4}\), we substitute back into the original equation \(\frac{13+2x}{4x+1} = \frac{3}{4}\).

• Calculate the numerator: \(13 + 2 \times \frac{49}{4}\) simplifies to \(39.5\).
• Calculate the denominator: \(4 \times \frac{49}{4} + 1\) simplifies to 50.
• The left side of the rewritten fraction is thus \(\frac{39.5}{50}\), which simplifies to \(\frac{3}{4}\).

Since the left-hand side equals the right-hand side of the original equation, your solution is verified to be correct.