Problem 57 - Prealgebra

Question

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

Step-by-Step Solution

Answer

When \(x = 3\), \(y = 0\).

Step 1: Substitute the Value of x into the Equation

Start with the equation given in the problem: \(4x + 3y = 12\). You are asked to find \(y\) when \(x = 3\). Substitute \(3\) for \(x\) in the equation. This gives us: \(4(3) + 3y = 12\).

Step 2: Simplify the Equation

Calculate the product \(4 \times 3\) which equals \(12\). The equation now becomes: \(12 + 3y = 12\).

Step 3: Isolate the Variable y

Subtract \(12\) from both sides of the equation to isolate the term containing \(y\). Thus, we have: \(3y = 0\).

Step 4: Solve for y

Divide both sides of the equation \(3y = 0\) by \(3\) to solve for \(y\). This results in \(y = 0\).

Key Concepts

Solving for y

When solving for a variable such as \(y\), the aim is to express \(y\) in terms of constants or other variables. In simpler terms, we want \(y\) all by itself on one side of the equation.
Here’s how to effectively solve for \(y\) following a series of algebraic steps:

• Start with your given equation, for example, \(3y + 12 = 24\).
• Isolate the term with \(y\) by performing operations to remove other numbers. Here, subtract \(12\) from each side to get \(3y = 12\).
• To solve for the single \(y\), divide each side by the coefficient in front of \(y\), which is \(3\) in this case. This results in \(y = 4\).

These steps simplify any equation and reveal the value of \(y\). Using methods like addition, subtraction, multiplication, and division helps untangle the equation pix so only \(y\) remains on one side.

Substitution Method

The substitution method is a powerful way to find the value of a variable when given an equation. This method involves replacing a variable with a known value to simplify the equation.
Here’s how you can apply it:

• Start with your equation, like \(4x + 3y = 12\).
• If you know a specific value for one of the variables, such as \(x = 3\), substitute this value directly into the equation. Thus, you replace \(x\) with \(3\), giving you \(4(3) + 3y = 12\).
• Complete the calculation so that the equation only contains one unknown variable, \(y\) in this case.

This step-by-step substitution reduces the complexity of the equation, turning it into one that’s easier to solve. It's a straightforward way to tackle problems involving linear equations.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations to make them easier to solve. It involves using operations like addition, subtraction, multiplication, and division to alter the form of an equation without changing its meaning.
To perform algebraic manipulation effectively, consider the following steps:

• Simplify each side of the equation as much as possible. This might mean distributing numbers or combining like terms.
• Move all terms with the variable to one side of the equation and constants to the other. For instance, if you have \(12 + 3y = 12\), subtract \(12\) from both sides to isolate the variable term, resulting in \(3y = 0\).
• Adjust the equation to solve for the variable. If you have \(3y = 0\), divide both sides by \(3\) to get \(y = 0\).

By methodically applying these techniques, you can change the structure of an equation to find solutions efficiently. Algebraic manipulation is at the heart of solving most linear equations and is key to untangling complex problems.