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Chapter 2: Problem 54

Use the addition property of equality to solve each equation. Check all solutions. $$1=y-5$$

Short Answer

Expert verified
The solution to the equation is \( y = 6 \)

Step by step solution

01

Identify the original equation

The original equation is \( 1 = y - 5 \)
02

Apply the addition property of equality

By adding 5 to both sides, the equation becomes \( 1 + 5 = y - 5 + 5 \)
03

Simplify

The equation simplifies to \( 6 = y \)
04

Check the solution

Substitute \( y = 6 \) back into the original equation to ensure it holds true. The left hand side becomes \( 1 = 6 - 5 \) which simplifies to \( 1 = 1 \), confirming the solution is correct.

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Key Concepts

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

Solving Linear Equations
The process of finding the value of the variable that makes a linear equation true is known as solving linear equations. These are equations where the variable, like y in our exercise, is raised to the first power. The goal is to isolate the variable on one side of the equation to find its value.

To solve the equation 1 = y - 5, we aim to get y by itself on one side. This involves reverse engineering the operations that have been applied to y. Since y has been decreased by 5, we counteract this by doing the opposite鈥攁dding 5 to both sides, which gives us the solution y = 6. Always perform the same operation on both sides of the equation to maintain the balance, just like a scale.
Algebraic Properties
Algebraic properties are the rules that govern the manipulation of algebraic expressions and equations. One of the foundational principles is the addition property of equality, which we just applied in our example.

This property states that if you add the same number to both sides of an equation, the equality is still valid. It鈥檚 like saying if two things are equal, and you give them the same amount more, they will remain equal. In our original problem, adding 5 to both sides utilizing this property enables us to keep the equation balanced and facilitate the solving process.
Equation Simplification
Simplification of an equation is a method of reducing it to its simplest form. That means combining like terms, eliminating parentheses, and making the equation as straightforward as possible鈥攊deally with the variable on one side and a number on the other.

In the given exercise, after applying the addition property of equality, we obtained the equation 1 + 5 = y - 5 + 5. Simplifying it by combining the numeric values on both sides gives us 6 = y. This process makes the solution more apparent and easily verifiable, leading to a clear, uncluttered result.
Solution Verification
Verifying the solution of an equation is crucial. It confirms whether the value obtained for the variable indeed satisfies the original equation. You perform a check by substituting the solution back into the initial equation and seeing if the equation holds true.

In our example, y = 6 is the solution we obtained. We verify this by replacing y with 6 in the original equation, leading to 1 = 6 - 5. Simplifying the right-hand side gives us 1 = 1, which is true, thereby confirming that our solution of y = 6 is correct. Always make solution verification a habit to ensure accuracy in your algebraic endeavors.

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Most popular questions from this chapter

To solve the inequality \(1 < \frac{1}{x},\) one student multiplies both sides by \(x\) to get \(x < 1 .\) Why is this not correct?

Solve each inequality and graph the solution on the number line. $$-4 < \frac{x-2}{2} < 6$$

Solve. The money \(m\) received from selling a house is given by the formula $$m=s-c$$ where \(s\) is the selling price and \(c\) is the agent's commission. Find the selling price of a house if the seller received \(? 217,000\) and the agent received \(\$ 13,020\).

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Solve each equation and round the result to the nearest tenth. $$\frac{-2.1(1.7 x+0.9)}{3.1}=-7.1(x-1.3)$$
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