Problem 64 - Prealgebra

Question

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

Step-by-Step Solution

Answer

When \(y = -4\), \(x = 6\).

Step 1: Substitute the Given Value for y

Start by substituting the given value of \(y = -4\) into the equation \(4x + 3y = 12\). This substitution transforms the equation into \(4x + 3(-4) = 12\).

Step 2: Simplify the Equation

Multiply 3 by -4 to simplify the equation: \(4x - 12 = 12\).

Step 3: Isolate the Term with x

Add 12 to both sides of the equation to isolate the term containing \(x\). This results in \(4x = 24\).

Step 4: Solve for x

Divide both sides of the equation by 4 to solve for \(x\). Thus, \(x = \frac{24}{4} = 6\).

Key Concepts

Linear Equations

A linear equation is a type of equation where the highest power of the variable is one. You can recognize them by their straightforward form, typically written as:

• \( ax + by = c \)

Where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. This form is known as the standard form of a linear equation. Each term is either a constant or the product of a constant and a single variable.
The beauty of linear equations lies in their simplicity. They describe straight lines in geometry when plotted on a graph. This is because the relationship between the variables is consistent and does not change.
Linear equations are foundational in algebra, providing the basis for understanding more complex equations and functions later on.

Solving for Variables

Solving for variables is all about finding the value of unknowns that make the equation true. When dealing with linear equations, the goal is to isolate the variable you are solving for, usually by using basic arithmetic operations like addition, subtraction, multiplication, or division.
When you solve for a variable, you're essentially "breaking down" the equation. You simplify it step by step until the variable stands alone on one side of the equation. This process is structured and logical, much like peeling the layers of an onion until you reach the core.
Let's consider our example: the equation was initially \(4x + 3y = 12\). If given \(y = -4\), we substitute first, and through arithmetic operations, we eventually find \(x = 6\). This final number is the "solution" for that particular case.
To ensure accuracy:

• Always perform the inverse operation to both sides of the equation
• Check your work by plugging the solution back into the original equation

Algebraic Substitution

Substitution is a method used in algebra to make calculations easier, especially when you have an equation with two variables. When you substitute, you're essentially replacing one variable with a given value or expression.
In the problem we solved, the technique involves directly replacing \(y\) with \(-4\) in the equation \(4x + 3y = 12\). This allows us to focus solely on finding the value of \(x\), simplifying the problem to a one-variable equation.
Substitution is incredibly valuable because:

• It reduces the complexity of the original equation
• Makes the equation easier to work with by focusing on just one unknown at a time
• Is a frequently used strategy in solving systems of equations

By substituting properly, you keep the problem straightforward while maintaining accuracy. It's a powerful tool often underestimated in its simplicity.