Problem 41 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$r+3.7=8$$

Step-by-Step Solution

Answer

The solution to the equation is \(r=4.3\)

Step 1: Apply the Addition Property of Equality

First, apply the addition property of equality to isolate r. This means subtracting 3.7 from both sides of the equation: \(r+3.7-3.7=8-3.7\)

Step 2: Simplify

Next, simplify both sides of the equation to isolate r on one side. This will result in the following: \(r=8-3.7\)

Step 3: Solve for r

Subtract to solve for the value of r: \(r=4.3\)

Step 4: Check your Solution

Check your solution by substituting r=4.3 into the original equation to verify if both sides equal. So if we replace r in the equation, we get \(4.3+3.7=8\), which is a true statement so the solution is correct.

Key Concepts

Solving Linear Equations

In algebra, solving linear equations is a fundamental skill that allows you to find the value of the unknown variable. A linear equation is an equation in which the highest power of the variable is one. The equation given in the exercise, r+3.7=8, is a simple example of a linear equation where r is the variable.

The goal when solving such equations is to isolate the variable on one side of the equation, which often requires applying inverse operations to both sides equally. For this particular problem, we aim to determine the value of r that makes the equation true. We do this by ensuring the variable stands alone, without any coefficients or added numbers, on one side of the equation.

Algebraic Properties

Algebraic properties are the rules that govern the operations and manipulations of algebraic expressions and equations. One of these essential properties is the addition property of equality, which states that adding or subtracting the same number from both sides of an equation will not change the equality.

This property is crucial in rearranging equations to solve for a variable. In the given problem, subtracting 3.7 from both sides allows you to maintain the balance of the equation while successfully isolating the variable r. Algebraic properties ensure that our mathematical operations are valid and that solutions obtained are accurate.

Equation Solving Steps

To solve an equation like r+3.7=8, we typically follow a series of equation solving steps. These steps are applied systematically to reach a solution in an organized and efficient manner.

• Step 1: Apply the Addition Property of Equality - Here, you perform the inverse operation to isolate the variable. By subtracting 3.7 from both sides, we 'cancel' it out on the left side.
• Step 2: Simplify - Once you apply the inverse operations, simplify the equation to isolate the variable further, which in this case simplifies to r=8-3.7.
• Step 3: Solve for the Variable - Perform the required calculation to find the value of the variable, resulting in r=4.3.

Following these steps methodically can help any student efficiently solve linear equations.

Solution Checking

Solution checking is a crucial last step in solving equations. It verifies that the value you've obtained satisfies the original equation. You do this by substituting the found value back into the original equation and ensuring that the left side equals the right side.

For the given problem, when we check our solution by replacing r with 4.3, we should perform the operation to confirm that 4.3+3.7=8. Since both sides equal 8, our solution is correct. By consistently checking solutions, you can be confident in the accuracy of your answers and understand that some equations may have multiple or no solutions.