Problem 50 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$6 z-3=z+2$$

Step-by-Step Solution

Answer

The solution for the equation is \(z = 1\).

Step 1: Rearrange the equation

In order to rearrange the equation, subtract \(z\) from both sides, obtaining \(5 z - 3 = 2\)

Step 2: Solve for z

Add 3 to both sides to isolate \(z\) resulting in \(5 z = 5\). From here, divide both sides by 5 to solve for \(z\), giving \(z = 1\)

Step 3: Verify the solution

Substitute \(z = 1\) back into the original equation \(6 z - 3 = z + 2\). If the two sides of the equation are equal, then the solution is correct. Substitute \(z = 1\), getting \(6*1 - 3 = 1 + 2\). After calculation, both sides equal to 3, so \(z = 1\) is the correct solution.

Key Concepts

Addition Property of Equality

When solving a linear equation, the addition property of equality allows us to maintain balance on both sides of the equation by adding the same number to each side. This concept is crucial because it means that the equality continues to hold true.
Consider the equation we are working on:
Before applying this property, we had the equation \[5z - 3 = 2\]To isolate the term with the variable, we added 3 to both sides. This resulted in the equation:\[5z = 5\]
By maintaining balance with the addition property, we ensure that the original equation's equality conditions are preserved. This step is necessary and prepares the equation for further simplification.

• Add the same quantity to both sides to keep the equation true.
• Helps to isolate the variable term.

Multiplication Property of Equality

The multiplication property of equality is another crucial tool when solving equations. This property allows us to maintain the balance of an equation by multiplying both sides by the same non-zero number.
Applying this property to our example equation:\[5z = 5\]We divided both sides by 5 to solve for \(z\). It's important to note that dividing is the same as multiplying by the reciprocal. Performing this action gives:\[z = 1\]
Using multiplication or division ensures balance, allowing us to find the variable's value while adhering to the principles of equality.

• Multiply or divide both sides by the same non-zero number.
• Need to keep the equation balanced.
• This helps to isolate the variable for the final step of solving the equation.

Verifying Solutions

After deriving a potential solution using properties of equality, it's critical to verify if the solution truly satisfies the original equation. Checking solutions not only confirms accuracy but also strengthens understanding of the process.
In our example, substituting \(z = 1\) back into the original equation:\[6z - 3 = z + 2\]Substitute and simplify:\[6 \times 1 - 3\]and\[1 + 2\]Both sides yield the result of 3, showing that they are indeed equal.
These steps confirm that \(z = 1\) is the correct solution.

• Substitute the solution back into the original equation.
• Simplify both sides to see if they are equal.
• Ensures the correctness of the solution.