Problem 25 - Algebra for College Students

Question

Solve each equation for \(y\). $$4 x+y=1$$

Step-by-Step Solution

Answer

y = 1 - 4x

Step 1: Isolate y

Starting with \(4x + y = 1\), subtract \(4x\) from both sides:
\(y = 1 - 4x\)

Step 2: State the solution

\(y = 1 - 4x\)

Key Concepts

Solving for y

To solve an equation for a specific variable, such as y, means to isolate that variable on one side of the equation. This way, we express y in terms of other variables or constants. First, identify the term that contains y. In our example, the original equation is: $$4x + y = 1.$$ Here, y is already on one side of the equation, but we need to isolate it completely by eliminating other terms on the same side.

Isolating Variables

Isolating a variable means moving all other terms to the opposite side of the equation. This often involves using basic arithmetic operations such as addition, subtraction, multiplication, or division.

In our example, to isolate y, we need to eliminate the term containing x, which is 4x. We do this by subtracting 4x from both sides of the equation. This action cancels out the 4x term on the left side: $$4x + y - 4x = 1 - 4x.$$

The equation simplifies to: $$y = 1 - 4x.$$ The term y is now isolated, which means y is expressed in terms of x and a constant.

Algebraic Manipulation

Algebraic manipulation involves rearranging an equation using various algebraic techniques. These operations follow specific rules, such as:

• Adding or subtracting the same quantity on both sides
• Using inverse operations to cancel out terms

Let's break down our example again. The original equation is: $$4x + y = 1.$$

We start by subtracting 4x from both sides of the equation to move the term containing x to the other side: $$4x + y - 4x = 1 - 4x.$$

This manipulation leaves us with: $$y = 1 - 4x.$$

Thus, we used algebraic manipulation to solve for y by isolating the variable and simplifying the equation.