Problem 30 - Introductory Algebra for College Students

Question

Solve each equation in using both the addition and multiplication properties of equality. Check proposed solutions. $$2 x+5=13$$

Step-by-Step Solution

Answer

After following the steps of isolating 'x', it is found that \(x = 4\) is the solution to the equation \(2x + 5 = 13\).

Step 1: Isolate the term 'x'

To isolate 'x', subtract 5 from both sides of the equation \(2x + 5 = 13\). This gives \(2x = 13 - 5\)

Step 2: Evaluation

Calculate the right side of the equation to obtain \(2x = 8\)

Step 3: Isolate 'x'

Now, to completely isolate 'x', divide both sides of the equation \(2x = 8\) by 2. This gives \(x = \frac{8}{2}\)

Step 4: Evaluation

Evaluate \(\frac{8}{2}\) to get \(x = 4\)

Step 5: Verification

To verify if \(x = 4\) is the correct solution, substitute 4 for 'x' in the original equation, leading to \(2*(4) + 5\), which simplifies to 8 + 5. This indeed equals 13, which is consistent with the original equation. Hence, the solution \(x = 4\) is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental concept used to solve equations. It states that if you add the same number to both sides of an equation, the equation remains balanced. This property helps keep the equality intact, and it's crucial when we're trying to isolate a variable.

In the given problem, we first subtracted 5 from both sides of the equation to begin isolating the variable. The equation was initially written as \(2x + 5 = 13\). Subtracting 5 is the direct application of the addition property of equality, as subtraction can be considered as adding a negative number. So, it became \(2x = 13 - 5\), simplifying further to \(2x = 8\).

Always remember:

• To maintain equality, perform the same operation on both sides.
• Use this property to strategically eliminate terms and simplify the equation.

Multiplication Property of Equality

The multiplication property of equality says that multiplying both sides of an equation by the same nonzero number does not change the equality. This property is essential for isolating the variable in equations.

In our exercise, once we got to the equation \(2x = 8\), we divided both sides by 2. This is an application of the multiplication property, as division is the inverse operation of multiplication. Essentially, you multiply both sides by the reciprocal of 2 (which is \(\frac{1}{2}\)). This results in the simpler equation \(x = \frac{8}{2}\), which we then solved to find \(x = 4\).

Key points to keep in mind:

• Always use nonzero numbers in the multiplication property.
• This property allows you to clear coefficients from variables to solve for them easily.
• Division is the same as multiplying by a reciprocal number in these contexts.

Verification of Solutions

Verification of solutions is a critical final step when solving equations. It ensures the solution you've found is correct and satisfies the original equation.

After determining that \(x = 4\) was the solution, we plugged this value back into the original equation \(2x + 5 = 13\). Substituting \(x = 4\), the equation became \(2(4) + 5\), which simplifies to \(8 + 5 = 13\). Since the left-hand side equaled the right-hand side of the original equation, our solution was verified.

To effectively verify solutions:

• Substitute the solution back into the original equation.
• Simplify and ensure both sides of the equation match.
• This step confirms correctness and boosts confidence in your solution.