Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each equation. Using the addition property of equality. Be sure to check your proposed solutions. $$-90+t=-35$$

Step-by-Step Solution

Answer

The solution to the equation is \( t = 55 \).

Step 1: Understand the Equation

Identify the equation which is \( -90+t=-35 \) in this case. Clearly, the variable to be solved for is \( t \).

Step 2: Apply the Addition Property of Equality

Isolate \( t \) by adding 90 to both sides of the equation to maintain the balance as availed by the addition property of equality. This gives us \( t = -35+90 \).

Step 3: Simplify the Equation

Evaluate the equation to find the value of \( t \). This gives \( t = 55 \).

Step 4: Check your solution

Substitute \( t \) with 55 in the original equation to check if both sides of the equation are equal. This gives us \( -90+55 = -35 \) which is true.

Key Concepts

Addition Property of Equality

Understanding the addition property of equality is essential for solving algebraic equations. This property tells us that you can add the same number to both sides of an equation without changing the equality. In the exercise \( -90+t=-35 \), when we add 90 to both sides, we are using this principle to adjust the equation in a way that isolates the variable:\
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• Original Equation: \( -90 + t = -35 \)\
• Add 90 to both sides: \( -90 + 90 + t = -35 + 90 \)\
• Simplified Equation: \( t = 55 \)\

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By applying this property, we maintain the balance of the equation, ensuring the two sides remain equal. A strong grasp of this concept allows students to confidently transform equations and bring them closer to the solution.

Isolate Variables

To solve for a variable means to isolate it on one side of the equation. Isolating the variable makes the equation clearer and prepares it for solving. Here's how isolation works in our exercise:\
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• Look at the equation and identify the variable and constant terms: \( -90+ t = -35 \)\
• Use the addition property to remove the constant term from the variable's side by adding 90 to both sides (as we've learned above).\
• With all constant terms on one side and the variable on the other, the variable \( t \) is successfully isolated: \( t = 55 \)\

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Isolating the variable is a fundamental skill in algebra. It simplifies the equation to a basic form where the variable—what you're solving for—stands alone.

Equation Simplification

Once a variable is isolated, the next step is often equation simplification. Simplification helps us identify the solution more clearly and confirm accuracy. To simplify the equation from our exercise:\
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• We start with the adjusted equation after isolation: \( t = -35 + 90 \)\
• Combine like terms by performing the addition: \( t = 55 \)\
• The equation is now simplified, and the solution for \( t \) is 55.\

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Equation simplification typically involves combining like terms, reducing fractions, or factoring. Its goal is to make the equation as straightforward as possible, often leading to a single step away from finding the solution.