Problem 87 $y-3=2(x+1)$... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 9: Problem 87

$y-3=2(x+1)$

Short Answer

Expert verified

y = 2x + 5

Step by step solution

Distribute the 2 on the right side

Start by expanding the right-hand side of the equation: \[ y - 3 = 2(x + 1) \]Distribute the 2 through the parenthesis: \[ y - 3 = 2x + 2 \]

Isolate y

Add 3 to both sides of the equation to isolate y: \[ y - 3 + 3 = 2x + 2 + 3 \]This simplifies to: \[ y = 2x + 5 \]

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Key Concepts

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

Distributive Property

The Distributive Property allows you to multiply each addend within a parenthesis by a factor outside the parenthesis.

For example, in the equation given: $ y - 3 = 2(x + 1) $, the right-hand side needs to be expanded using the Distributive Property. Here, we distribute the 2 to both x and 1: \[ y - 3 = 2 \times x + 2 \times 1 \] Which simplifies to: \[ y - 3 = 2x + 2 \]

The Distributive Property helps to simplify the equation, making it easier to solve for the variable of interest. Always apply this property to eliminate parentheses so you can work with simpler expressions.

Isolating Variables

Isolating the variable means getting the variable alone on one side of the equation.

In order to isolate y in the equation $ y - 3 = 2x + 2 $, you need to perform operations to undo any additions, subtractions, multiplications, or divisions attached to y.

Here, y is subtracted by 3. To reverse this, you need to add 3 to both sides of the equation: \[ y - 3 + 3 = 2x + 2 + 3 \] Which simplifies to: \[ y = 2x + 5 \]

This process leaves y isolated on the left side of the equation while keeping the equation balanced. Knowing how to isolate variables is crucial for solving all types of equations.

Linear Equations

A linear equation is an equation that makes a straight line when it is graphed. The general form of a linear equation in two variables is $ y = mx + b $, where m is the slope and b is the y-intercept.

In the exercise provided, we end up with the linear equation: \[ y = 2x + 5 \] Here, 2 is the slope (it tells us how steep the line is) and 5 is the y-intercept (it tells us where the line crosses the y-axis). Understanding the structure of linear equations helps in identifying key characteristics of their graphs.

By mastering the principles of solving linear equations, you unlock the ability to solve a wide variety of real-world problems that can be modeled algebraically.

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91影视

\(y-3=2(x+1)\)

Short Answer

Step by step solution

Distribute the 2 on the right side

Isolate y

Key Concepts

Distributive Property

Isolating Variables

Linear Equations

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