Problem 59 - Prealgebra

Question

Suppose \(4 x+3 y=12 .\) Find \(y\) if: $$x=-\frac{1}{4}$$

Step-by-Step Solution

Answer

If \(x = -\frac{1}{4}\), then \(y = \frac{13}{3}\).

Step 1: Substitute the value of x into the equation

The given equation is \(4x + 3y = 12\). You need to plug in \(x = -\frac{1}{4}\) into this equation, replacing \(x\). Then the equation becomes \(4\left(-\frac{1}{4}\right) + 3y = 12\).

Step 2: Simplify the equation

Simplify the term involving \(x\). Compute \(4 \times -\frac{1}{4} = -1\). Now the equation is \(-1 + 3y = 12\).

Step 3: Solve for y

To isolate \(y\), add 1 to both sides of the equation: \(-1 + 3y + 1 = 12 + 1\), which simplifies to \(3y = 13\).

Step 4: Divide to find y

Divide both sides of the equation by 3 to solve for \(y\): \(y = \frac{13}{3}\).

Key Concepts

Substitution Method

The substitution method is used to solve systems of equations. In this method, we replace one variable with an expression that is equivalent in terms of another variable.
This strategy helps to simplify the equation, making it easier to solve.

• Consider the original problem: Let's solve for \(y\) in the equation \(4x + 3y = 12\) given that \(x = -\frac{1}{4}\).
• Substitute \(x\) in the equation: Replace \(x\) with the given value, creating a simpler equation with just one variable.
• The equation then reads \(4(-\frac{1}{4}) + 3y = 12\).

The substitution method is a powerful tool, especially when you have one value of a variable and need to find another. It's often paired with simplification to achieve a clear view of the solution.

Simplifying Equations

Simplifying an equation is about reducing it to its most basic form so that it becomes easy to manage and solve.
This involves combining like terms and performing basic arithmetic.

• After substituting \(x = -\frac{1}{4}\), we have \(4(-\frac{1}{4}) + 3y = 12\).
• Next, compute \(4 \times -\frac{1}{4}\), which simplifies to \(-1\).
• Substitute this back to get \(-1 + 3y = 12\).

This streamlined equation makes it clear that two variables have been consolidated, making it ready for the next stage: solving for \(y\). Simplifying plays a crucial role in keeping the math manageable and ensuring fewer mistakes.

Solving for a Variable

Solving for a variable means isolating it on one side of the equation. This lets you determine its value based solely on numbers or other variables.

• From the simplified equation \(-1 + 3y = 12\), we want \(y\) alone.
• Add 1 to both sides to balance the equation: \(-1 + 1 + 3y = 12 + 1\).
• This simplifies to \(3y = 13\).
• Finally, divide each side by 3 to find \(y\): \(y = \frac{13}{3}\).

Now you know \(y\)'s value. Solving for a variable requires clear operations that maintain balance across the equation. This ensures the solution is accurate and verifiable.