Problem 23 - Algebra 1

Question

Solve the equation. $$3-\frac{3}{4} x=-6$$

Step-by-Step Solution

Answer

The solution to the equation \(3 - \frac{3}{4}x = -6\) is \(x = 12\).

Step 1: Simplify and Rearrange the Equation

First, the equation \(3-\frac{3}{4}x=-6\) can be rearranged to \(-\frac{3}{4}x=-6-3\). This results in \(-\frac{3}{4}x=-9\).

Step 2: Remove the Fraction

To remove the fraction, multiply both sides of the equation by the reciprocal of \(-\frac{3}{4}\), which is \(-\frac{4}{3}\). Therefore, the equation becomes: \(x = 9 \times -\frac{4}{3}\).

Step 3: Solve for x

After multiplication, it results in \(x=12\).

Key Concepts

Linear Equation

In mathematics, the term linear equation refers to an equation that forms a straight line when graphed on a coordinate plane. Such equations typically look like \( ax + b = c \), where \(a, b,\) and \(c\) are constants and \(x\) is the variable we want to solve for. The power of \(x\) is always 1, which means there are no \(x^2\) terms or higher powers of the variable present. The solution to a linear equation is the value of \(x\) that makes the equation true, essentially finding the point where the line crosses the \(x\)-axis, known as the x-intercept. In the given exercise, \(3 - \frac{3}{4} x = -6\) is a linear equation with \(a = -\frac{3}{4}\), \(b = 3\), and \(c = -6\). Solving this equation involves isolating the variable \(x\) to find the point where the line \(y = 3 - \frac{3}{4} x\) intersects with the \(x\)-axis.

Algebraic Fractions

An algebraic fraction is a fraction that contains a variable, such as \(\frac{3}{x}\) or \(\frac{a + b}{c + d}\). Algebraic fractions can appear intimidating, but they are manipulated using the same principles as numerical fractions. To solve an equation with algebraic fractions, we often find a common denominator to combine terms, cross-multiply to clear the fractions, or use the reciprocal of a fraction to eliminate it. In practice, as in our exercise, to isolate the variable and solve the equation \(-\frac{3}{4}x = -9\), we need to 'get rid of' the fraction by multiplying by its reciprocal.

Reciprocal in Algebra

The reciprocal of a number or an algebraic expression is 1 divided by that number or expression. It is essentially flipping the numerator and denominator of a fraction. In algebra, using reciprocals is a powerful way to simplify equations, particularly when trying to eliminate fractions. When you multiply a number by its reciprocal, the result is 1. Therefore, when an equation contains a fraction, multiplying both sides of the equation by the reciprocal of that fraction will remove the fraction. Applying this to our exercise, the reciprocal of \(-\frac{3}{4}\) is \(-\frac{4}{3}\). Multiplying both sides of \(-\frac{3}{4}x = -9\) by \(-\frac{4}{3}\) results in \(x = 12\), effectively eliminating the fraction.

Isolation of Variables

The process of isolation of variables involves manipulating an equation to get the variable of interest by itself on one side of the equation. This is the crux of solving any algebraic equation. By using a series of algebraic operations, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same amount, we can isolate the variable. The goal is to have the variable on one side and a numerical value on the other, showing the solution. For the given problem, isolating the variable \(x\) involved clearing the algebraic fraction and then multiplying to find the value of \(x\) that satisfies the original equation.