Problem 67 - Prealgebra

Question

Suppose \(4 x+3 y=12 .\) Find \(x\) if: $$y=0$$

Step-by-Step Solution

Answer

\(x = 3\) when \(y = 0\).

Step 1: Substitute the Value of y

Given the equation \(4x + 3y = 12\) and \(y = 0\). We substitute \(0\) for \(y\) in the equation: \[4x + 3(0) = 12\].

Step 2: Simplify the Equation

Simplify the equation from step 1: \[4x = 12\].

Step 3: Solve for x

To isolate \(x\), divide both sides of the equation \(4x = 12\) by 4: \[x = \frac{12}{4}\].

Step 4: Simplify the Solution

Simplify \(x = \frac{12}{4}\) to get \(x = 3\).

Key Concepts

Solving for a Variable

In algebra, solving for a variable means finding the value of an unknown that makes the equation true. Think of it as figuring out what number, when substituted back into the equation, keeps both sides equal.

For example, if we have the equation \(4x + 3y = 12\) and we know that \(y = 0\), our goal is to find out what value of \(x\) will satisfy this equation. Solving for a variable usually involves rearranging the equation to get the variable of interest by itself on one side.

• Start with the original equation.
• Identify which variable you need to solve for.
• Use algebraic operations to isolate this variable on one side of the equation.

In this exercise, we correctly identify that \(y\) is given as \(0\), and we proceed through substitution and simplification to find that \(x = 3\). By mastering these steps, you'll be able to tackle any linear equation with confidence!

Substitution Method

The substitution method involves substituting a specified value or expression for a variable in an equation. It's a strategic approach, especially useful when you are given the value of one variable and need to find another.

Consider the problem where you're given \(4x + 3y = 12\) and \(y = 0\). By substituting \(0\) for \(y\) in the equation, the equation becomes much simpler: \[4x + 3(0) = 12\]

• Identify the given value or expression for a variable (\(y=0\) here).
• Substitute this value into the equation.
• Solve the resulting simpler equation.

By applying this method, we seamlessly transition from a two-variable equation to a one-variable equation, making it far easier to solve.

Simplifying Equations

Simplifying equations is a crucial skill in algebra, helping to make equations easier to work with by reducing complexity. It primarily involves combining like terms and performing basic arithmetic operations.

Take the equation \(4x + 3(0) = 12\) from the substitution step. Since multiplying any number by zero results in zero, \(3(0)\) simplifies to \(0\). The equation rapidly simplifies to \(4x = 12\).

• Remove any terms that equal zero.
• Combine like terms where possible.
• Ensure both sides of the equation are balanced.

The next logical step after simplifying is isolating the desired variable, leading you directly to the solution, \(x = 3\). Mastering simplification makes solving equations less daunting and more intuitive.