Problem 66 - Prealgebra

Question

Suppose \(4 x+3 y=12 .\) Find \(x\) if: $$y=\frac{5}{3}$$

Step-by-Step Solution

Answer

The value of \(x\) is \(\frac{7}{4}\).

Step 1: Substitute the given value of y into the equation

The original equation is \(4x + 3y = 12\). We are given that \(y = \frac{5}{3}\). Substitute this value into the equation to replace \(y\). The equation becomes: \(4x + 3\left(\frac{5}{3}\right) = 12\).

Step 2: Simplify the equation

Simplify the terms inside the equation. Multiply the constants: \(3 \times \frac{5}{3} = 5\). The equation now becomes \(4x + 5 = 12\).

Step 3: Isolate the variable x

To find the value of \(x\), isolate it on one side of the equation. Subtract 5 from both sides of the equation: \(4x + 5 - 5 = 12 - 5\). This simplifies to \(4x = 7\).

Step 4: Solve for x

Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{7}{4}\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in solving linear equations, and it's especially useful when we know the value of one variable. In simple terms, substitution involves replacing a variable with a known value to simplify the equation. This is exactly what happened in our exercise where we knew the value of \( y \) and substituted it into the equation.

• Identify the variable with a known value, in this case \( y = \frac{5}{3} \).
• Replace the variable in the equation with this known value: \( 4x + 3 \left( \frac{5}{3} \right) = 12 \).

By replacing \( y \) with its known value, we transform our equation into one with a single variable \( x \) that we can now solve with greater ease.

Simplifying Equations

Simplifying equations involves reducing them to a form that is easier to solve. This often includes performing arithmetic operations to simplify terms. In our example, after substituting the value of \( y \), the equation became \( 4x + 3 \left( \frac{5}{3} \right) = 12 \). The next step was to simplify the expression within the equation.
When you see an expression like \( 3 \left( \frac{5}{3} \right) \), you should:

• Multiply the coefficient outside the parentheses by the fractional value inside: \( 3 \times \frac{5}{3} = 5 \).
• Simplify the equation by replacing \( 3 \left( \frac{5}{3} \right) \) with 5, resulting in a simpler equation of \( 4x + 5 = 12 \).

These steps make the equation much simpler and bring you closer to isolating the variable.

Isolating Variables

Isolating the variable is the process of manipulating the equation to have the variable on one side and everything else on the other. Once we've simplified our equation, \( 4x + 5 = 12 \), the goal becomes getting \( x \) by itself.
To isolate \( x \):

• Subtract 5 from both sides of the equation to get \( 4x = 12 - 5 \), which simplifies to \( 4x = 7 \).
• Divide both sides of the equation by 4 to solve for \( x \): \( x = \frac{7}{4} \).

By isolating \( x \), we've effectively solved the equation and found that \( x = \frac{7}{4} \). This method works the same way for any linear equation, allowing you to find the value of the unknown variable.