Problem 60 - Prealgebra

Question

Suppose \(4 x+3 y=12 .\) Find \(y\) if: $$x=\frac{3}{2}$$

Step-by-Step Solution

Answer

When \(x = \frac{3}{2}\), \(y\) equals 2.

Step 1: Substitute the Given Value of x

We are given the equation \(4x + 3y = 12\) and the specific value of \(x = \frac{3}{2}\). Substitute \(\frac{3}{2}\) for \(x\) in the equation: \(4 \left(\frac{3}{2}\right) + 3y = 12\).

Step 2: Simplify the Equation

First, calculate \(4 \times \frac{3}{2}\). This simplifies to \(2 \times 3 = 6\). Now, the equation becomes \(6 + 3y = 12\).

Step 3: Isolate the Variable y

Subtract 6 from both sides of the equation to isolate the term with \(y\). This gives us: \(3y = 12 - 6\), which simplifies to \(3y = 6\).

Step 4: Solve for y

Divide both sides of the equation by 3 to solve for \(y\). We get: \(y = \frac{6}{3}\), which simplifies to \(y = 2\).

Key Concepts

Solving for y

When you encounter a linear equation like \(4x + 3y = 12\) and need to solve for \(y\), the goal is to isolate \(y\) on one side of the equation. This makes it simple to see what value \(y\) holds. To do this, you perform a series of steps using basic algebraic operations. For example, given \(x = \frac{3}{2}\), you start by substituting this value into the equation, which then requires simplifying and manipulating the equation until \(y\) stands alone on one side, fully revealed. This might involve subtracting or adding numbers and, ultimately, dividing by the coefficient in front of \(y\) to get the exact value.

Substitution method

The substitution method is a technique often used to solve a system of equations, but it can also be applied on single equations when specific values for variables are given. In this instance, for the equation \(4x + 3y = 12\), substituting \(x = \frac{3}{2}\) into the equation helps turn a two-variable equation into a one-variable equation. Here’s what happens: replacing \(x\) with \(\frac{3}{2}\) directly affects the equation structure, simplifying it into one that we can easily solve for \(y\). It transforms the exercise from dealing with multiple unknowns to only focusing on \(y\), streamlining the process significantly.

Basic algebraic operations

Basic algebraic operations include addition, subtraction, multiplication, and division. These operations are essential tools in solving linear equations. In our given example of \(4x + 3y = 12\), you first simplify the equation by multiplying and then subtracting values to make solving easier. Specifically, once \(4 \times \frac{3}{2} = 6\) is computed, the equation becomes \(6 + 3y = 12\). Subtract 6 from both sides to isolate the terms with \(y\), followed by dividing both sides of \(3y = 6\) by 3 to solve for \(y\). These steps use simple algebraic rules, illustrating how mathematical operations can methodically simplify the expression.