Problem 61 - Prealgebra

Question

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

Step-by-Step Solution

Answer

3y = 12, so y = 4.

Step 1: Understanding the Given Equation

The problem gives us a linear equation in two variables: \(4x + 3y = 12\). We want to find the value of \(y\) when \(x = 0\).

Step 2: Write the equation in standard form

Rearrange and simplify the equation.

Step 3: Apply the solution method

Use factoring, quadratic formula, substitution, or other methods.

Step 4: Verify the solution(s)

Check solutions in the original equation.

Step 5: State the final answer

List all valid solutions.

Key Concepts

Solving for y

When working with equations, solving for a specific variable means isolating that variable on one side of the equation. In this exercise, we are asked to solve for the value of \(y\). Our given equation is \(4x + 3y = 12\). Our goal is to isolate \(y\) by making sure it appears alone on one side of the equation.

To achieve this, we will first substitute the given value of \(x = 0\) into the equation, which simplifies our task. By substituting, the equation transforms into \(3y = 12\), as the term \(4x\) becomes 0 when \(x = 0\). Next, we isolate \(y\) by dividing both sides of the equation by 3, giving us \(y = \frac{12}{3}\). Simplifying this, we find \(y = 4\).

• Identify the variable you need to solve for.
• Re-arrange the equation to isolate this variable.
• Simplify the expression to find the variable's value.

Two-Variable Equations

The equation \(4x + 3y = 12\) is a linear equation with two variables, \(x\) and \(y\). These equations typically form straight lines when graphed on a coordinate plane. A key concept in working with these equations is understanding how to solve for one variable when a specific value for the other is provided. This involves manipulating the equation algebraically.

Linear equations in two variables can represent numerous solutions, each corresponding to a point on the line they form. When we substitute a value for one variable, we effectively pinpoint one unique solution from among these possibilities. In real-world applications, such equations are used to model relationships between two different quantities.

• Two-variable equations consist of two unknowns, usually \(x\) and \(y\).
• They are graphically represented as straight lines in the coordinate system.
• Solving them involves substituting known values to find unknown variables.

Substituting Values

Substitution is an essential technique used to simplify equations and make solving them easier. When given an equation with two variables, one can find the value of one variable by substituting a specific value for the other.

In this exercise, we started with the equation \(4x + 3y = 12\) and the information that \(x = 0\). By substituting \(x = 0\) into the equation, we eliminate \(x\) from the equation, simplifying it to \(3y = 12\). This transforms a two-variable problem into a straightforward single variable equation, making it much simpler to solve.

• Identify which variable to substitute into the equation.
• Replace the variable with the given value.
• Solve the resulting equation to find the value of the other variable.