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Chapter 2: Problem 105

Solve each equation for \(y\) $$ 5 x-y=2 $$

Short Answer

Expert verified
y = 5x - 2

Step by step solution

01

Move terms involving y to one side

Start with the equation: 5x - y = 2 To isolate the term with y, move it to the left side by subtracting 5x from both sides of the equation: 5x - y - 5x = 2 - 5x This simplifies to: -y = -5x + 2
02

Solve for y

To isolate y, multiply both sides of the equation by -1: (-1)(-y) = (-1)(-5x + 2) This results in: y = 5x - 2

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Understanding how to isolate variables is crucial when solving linear equations. When we isolate a variable, we are trying to get that variable alone on one side of the equation. This often involves moving terms from one side of the equation to the other using basic arithmetic operations.

In the exercise, we isolated the variable y in two main steps:
  • First, we moved the term involving y to one side of the equation by subtracting 5x from both sides. This helped us focus on y by getting rid of the x term on the original side.

  • Next, we handled the coefficient of y. Multiplying both sides by -1 allowed us to transform -y into y. By doing these steps, we successfully isolated y.
These are the basic principles of isolating a variable, which you'll frequently use in algebra.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging equations to solve for a particular variable. This involves using operations like addition, subtraction, multiplication, and division.

In the provided solution, we used subtraction to move 5x to the other side of the equation. Remember, whatever you do to one side of the equation, you must do to the other.

Here's a breakdown:
  • Starting equation: 5x - y = 2
  • Subtract 5x from both sides: -y = -5x + 2

Then, we multiplied both sides by -1 to simplify:
  • Multiply by -1: -y x -1 = (-5x + 2) x -1
  • This results in: y = 5x - 2

Practicing these manipulations will make your solving strategies more effective.
Linear Equations
Linear equations are algebraic equations where each term is either a constant or the product of a constant and a single variable. These equations graph as straight lines in a coordinate plane.

The general form is: Ax + By = C, where A, B, and C are constants.

In the given problem, the equation 5x - y = 2 is already in a near-standard form. We merely solved for y to express this as: y = 5x - 2.

This equation now tells us the relationship between y and x. For every increase in x, y increases by 5 times the value of x, minus 2. Understanding the format and solution of linear equations is fundamental in algebra.skills will prove useful in many math and real-world applications.

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