Problem 39 - Introductory Algebra for College Students

Question

Solve each equation using the addition property of equality. Be sure to check your proposed solutions. $$830+y=520$$

Step-by-Step Solution

Answer

The solution to the equation \( 830 + y = 520 \) is \( y = -310 \).

Step 1: Re-arrange equation

Re-arrange the equation to find the value of 'y'. The given equation is \( 830 + y = 520 \). We can rewrite this equation as \( y = 520 - 830 \) by subtracting 830 from each side of the equation. The purpose of this step is to get 'y' alone on one side of the equality.

Step 2: Compute 'y'

Next, compute the value of 'y'. Doing the arithmetic will give us \( y = -310 \). This value is what 'y' must be in order for the equation to hold true.

Step 3: Check the solution

Substitute \( y = -310 \) into the original equation \( 830 + y = 520 \) to see if the left side equals the right side. Doing so gives us \( 830 - 310 = 520 \), which is true. This confirmation that the original equation holds true with this value of 'y' shows that the solution is correct.

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental principle in algebra that states if you add the same amount to both sides of an equation, the equation remains balanced and the equality is maintained.

For example, consider the equation provided in the exercise, where you have to solve for 'y' in the equation \(830 + y = 520\). To isolate 'y', you deduct 830 from both sides, applying the addition property of equality. What this does is essentially balance the act of removing 830 units from one side by also removing it from the other, keeping the equality in check. The altered equation then becomes \(y = 520 - 830\), which simplifies down to \(y = -310\). This process ensures that the solution remains valid and the integrity of the equation is upheld.

Algebraic Problem Solving

Algebraic problem solving involves a systematic approach to finding the values of unknowns within equations. It is driven by applying algebraic operations and properties logically and sequentially to transform the original equation into a simpler form.

The step-by-step solution to the given equation demonstrates this systemic approach: Starting with re-arrangement to facilitate easier operations and concluding with arithmetic computation. The goal is to modify the original equation to progressively isolate the variable of interest. The steps highlight the importance of understanding the rules and properties of algebra which serve as tools in the problem-solving process to achieve the desired outcome.

Checking Algebraic Solutions

Checking algebraic solutions is an essential part of the problem-solving process because it verifies the accuracy of the found solutions.

After computing that \(y = -310\), it is crucial to test this solution by substituting back into the original equation. In our case, when we replace 'y' with -310 in the equation \(830 + y = 520\), we get \(830 - 310 = 520\), which confirms that the both sides are equal and thus validates our solution. This practice not only confirms the correctness of our answer but also reinforces our understanding of the relationships within the equation.

Isolating Variables

Isolating variables is a strategy used to solve for unknowns in equation form. The aim is to get the variable alone on one side of the equation, ideally resulting in a statement like \(x = [number]\) or \(y = [number]\).

In the provided exercise, isolating 'y' was achieved by removing the number 830 from the same side as 'y'. This act of transferring a number from one side of an equation to the other, while maintaining equality, is the centerpiece of solving linear equations. By isolating the variable, we make the equation's statement clear and directly solvable, leading us to the specific value that the variable represents in the context of the equation.