Problem 58 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ y-3=-7 $$

Step-by-Step Solution

Answer

The solution is \( y = -4 \).

Step 1: Understanding the Equation

The problem presents a linear equation in the form of \( y - 3 = -7 \). Our goal is to find the value of \( y \) that makes this equation true.

Step 2: Isolating the Variable

To solve for \( y \), we need to isolate it on one side of the equation. Currently, \( y \) is accompanied by \(-3\). To remove \(-3\), we need to add \(3\) to both sides of the equation.

Step 3: Performing the Addition

Add \(3\) to both sides of the equation: \((y - 3) + 3 = -7 + 3\) Simplifying both sides gives us: \(y = -4\)

Step 4: Verification

Let's verify our solution by substituting \( y = -4 \) back into the original equation: \(y - 3 = -7\) becomes \((-4) - 3 = -7\). This simplifies to \(-7 = -7\), confirming our solution is correct.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of an unknown variable that makes the equation true. In the given equation, \( y - 3 = -7 \), our objective is to find the value of \( y \). Linear equations are equations where the variable has an exponent of one. They usually look something like \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable.

Here, the equation is simple with minimal terms: \( y \) is the variable we're trying to isolate. This means you'll need to remove any other numbers or operations on the same side of \( y \). Solving involves using basic arithmetic operations like addition, subtraction, multiplication, or division to simplify both sides of the equation until you isolate the variable completely on one side. By doing so, you find the specific value that \( y \) must take for the equation to be valid.

Isolation of Variables

Isolation of variables is an essential technique in solving equations. The primary goal is to have the variable by itself on one side of the equation. In our example, the equation is \( y - 3 = -7 \). Here, \( y \) is accompanied by \(-3\), which we need to eliminate to solve for \( y \).

The rule to remember is that you can do anything to one side of the equation as long as you do the same to the other side. To isolate \( y \), add \( 3 \) to both sides of the equation. This operation simplifies to:

• Original: \( y - 3 \)
• Add \( 3 \) on both sides: \((y - 3) + 3 = -7 + 3\)

After performing the addition, you are left with \( y = -4 \). At this stage, \( y \) is by itself on one side of the equation, achieving our goal of isolation. This technique is crucial as it helps us precisely solve for unknowns in algebraic problems.

Verification of Solutions

Verification of solutions ensures that the value obtained for the variable is correct. After finding the potential solution, you must substitute it back into the original equation to check all computations are accurate. If both sides of the equation are equal after substituting, your solution is correct.

In the exercise, after solving \( y = -4 \), we substitute it back into the original equation \( y - 3 = -7 \) to verify:

• Substitute \( y = -4 \) into \( y - 3 = -7 \)
• Calculate \((-4) - 3 = -7 \)

This simplifies to \(-7 = -7\), which confirms our solution is accurate. This verification step is critical in problem-solving processes because it strengthens your confidence in the accuracy of your results. Always verify your solutions to ensure you have not made any mistakes during the calculation.