Problem 68 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \(x\) is \(5.25\).

Step 1: Substitute the value of y

Given the equation is \(4x + 3y = 12\), substitute \(y = -3\) into the equation. This gives us \(4x + 3(-3) = 12\). Simplify this to \(4x - 9 = 12\).

Step 2: Isolate the term with x

To isolate the term containing \(x\), add 9 to both sides of the equation. This gives \(4x = 12 + 9\). Simplifying the right-hand side gives \(4x = 21\).

Step 3: Solve for x

Divide both sides of the equation by 4 to solve for \(x\). This gives \(x = \frac{21}{4}\). Simplify if necessary to find \(x = 5.25\).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in solving systems of linear equations. It involves replacing one variable with an expression involving the other variable to simplify the solving process. Consider the given equation:

• Equation: \(4x + 3y = 12\)
• Substituted Value: \(y = -3\)

By substituting the value of \(y\) directly into the equation, we streamline our calculation process. The equation transforms immediately, allowing for simpler arithmetic operations. In this specific problem, substituting \(y = -3\) converts the original equation into \(4x - 9 = 12\). As you can see, the substitution reduces complexity, providing a straightforward path to isolating variables. This method is extremely helpful for linear equations where one variable’s value can be readily substituted without ambiguity. It ensures that our solving steps remain clear and manageable.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations like our own here. It provides the theoretical foundation to study and handle vector equations efficiently. In solving equations such as \(4x + 3y = 12\), linear algebra helps us understand the relationship between variables, and how they can be manipulated to find solutions. Linear equations take a straight-line form, where variables are raised to the first power. They often involve expressions that can be represented in matrices or simple line equations. By frequently practicing solving such equations, one develops a solid grasp of linear functions, which is essential for advanced mathematical studies. By exploring multiple ways to handle such equations, like the substitution method, learners gain versatility in their approach.

Isolating Variables

Isolating variables is a crucial step in solving algebraic equations. It refers to the process of manipulating the equation to have one variable all by itself on one side of the equation. For example, in our problem, after substituting \(y = -3\) into the equation, we're left with \(4x - 9 = 12\).

• Step 1: Add 9 to both sides to remove any non-variable terms. This gives us \(4x = 21\).
• Step 2: Isolate \(x\) by dividing each side by 4 to yield \(x = \frac{21}{4}\).

This simplified approach is widely applicable across different types of algebraic problems. Isolating the variable is not just a step in a problem, but a fundamental skill in algebra. It allows us to solve equations accurately while providing a deeper understanding of the relations among variables within any given system. As learners practice isolating variables, they greatly enhance their problem-solving skills and readiness for more complex equations.