Problem 14 - Precalculus

Question

Solve each equation. $$\frac{x-1}{4}+\frac{2 x+3}{-1}=0$$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{13}{7} \).

Step 1: Simplify the equation

The given equation is \( \frac{x-1}{4} + \frac{2x+3}{-1} = 0 \). First, simplify the fractions. The second fraction can be simplified as follows: \( \frac{2x+3}{-1} = -(2x + 3) \). Substitute back to get: \( \frac{x-1}{4} - (2x + 3) = 0 \).

Step 2: Remove the fractions

Multiply both sides of the equation by 4 to eliminate the fraction. This gives: \( x - 1 - 4(2x + 3) = 0 \).

Step 3: Distribute and combine like terms

Distribute the \(4\) in the equation: \( x - 1 - 8x - 12 = 0 \). Now, combine the like terms: \( x - 8x - 1 - 12 = 0 \). This simplifies to \( -7x - 13 = 0 \).

Step 4: Solve for x

Add 13 to both sides to isolate the term with \(x\): \( -7x = 13 \). Now divide both sides by \(-7\) to solve for \(x\): \( x = -\frac{13}{7} \).

Key Concepts

Fraction Simplification

Fraction simplification is an essential step when solving linear equations involving fractions. It simplifies the equation, making it easier to work with. When you have a fraction like \( \frac{2x+3}{-1} \), you can simplify it by considering the negative sign in the denominator. In this case, dividing by \(-1\) is equivalent to multiplying by \(-1\). This means that \( \frac{2x+3}{-1} \) becomes \(-(2x+3)\).
Simplification is critical because it reduces complexity and allows you to apply further algebraic techniques more easily. For fractions that can be simplified, look for common factors in the numerator and the denominator that can be divided out.
A helpful tip is to always keep an eye on negative signs hidden in denominators since they can simplify how terms interact within the equation. Once simplified, substitute back into the equation to continue solving.

Combining Like Terms

Combining like terms is a method used to simplify an equation by reducing the number of terms. Like terms contain the same variable raised to the same power. For example, in the equation \(x - 8x - 1 - 12\), the terms \(x\) and \(-8x\) are like terms because they both contain the variable \(x\).
To combine them, simply add or subtract their coefficients: \(1x - 8x = -7x\). Also, combine the constant terms \(-1\) and \(-12\), which gives you \(-13\).
After combining like terms, the equation \(x - 8x - 1 - 12 = 0\) transforms into \(-7x - 13 = 0\). This process significantly simplifies the equation, making it easier to isolate the variable and solve for it.

• Look for terms with matching variables.
• Combine their coefficients by performing addition or subtraction.
• Always notice terms without variables (constants) and group them together.

Eliminating Fractions

Eliminating fractions is a useful step to simplify equations, allowing you to focus on integer coefficients and constants. In the problem, we had the fraction \(\frac{x-1}{4}\).
To eliminate it, you can multiply each term in the equation by the denominator, which is \(4\) in this case. This action applies across all terms, making the equation easier to manage.
When multiplied by \(4\), the fraction becomes an integer: \(x-1\), and the rest of the equation scales proportionally. The equation \( \frac{x-1}{4} - (2x + 3) = 0 \) becomes \( x - 1 - 4(2x + 3) = 0 \).
Eliminating fractions simplifies the equation and reduces errors during calculation, especially if there's more than one fraction.

• Identify the least common denominator when multiple fractions are present.
• Multiply all terms in the equation by this common denominator.
• Rewrite the equation using the resulting simplified terms.

Negative Coefficients in Equations

Negative coefficients occur quite frequently in algebra, and it is essential to handle them correctly while solving equations. In the step where we simplified the equation to \(-7x - 13 = 0\), the coefficient of \(x\) is \(-7\).
To isolate the variable \(x\), perform operations that will eliminate negative signs, moving towards a solution where \(x\) is positive.
Start by isolating the term with \(x\): add 13 to both sides, resulting in \(-7x = 13\).
Then divide both sides by \(-7\), giving \(x = -\frac{13}{7}\). Working through negative coefficients requires careful attention to sign changes during multiplication or division to ensure accuracy.

• When adding or subtracting, maintain proper signs on all terms.
• If dividing or multiplying by a negative number, remember to flip inequality signs if applicable (not needed here).
• Keep the goal in sight: simplify until the variable is isolated with a positive coefficient if possible.