Problem 61 - Prealgebra

Question

Use the formula \(y=\frac{1}{2} x-3\) to find \(y\) if: $$x=2$$

Step-by-Step Solution

Answer

For \(x = 2\), \(y = -2\).

Step 1: Identify the given values

We are given the equation \(y=\frac{1}{2}x - 3\) and the value of \(x\) which is 2. We need to substitute \(x = 2\) into this equation to find the value of \(y\).

Step 2: Substitute the known value into the equation

Substitute \(x = 2\) into the equation: \[ y = \frac{1}{2}(2) - 3 \]

Step 3: Simplify the equation

First, calculate \(\frac{1}{2} \times 2\):\[ \frac{1}{2} \times 2 = 1 \]Now, substitute back into the equation:\[ y = 1 - 3 \]

Step 4: Solve for y

Subtract 3 from 1 to solve for \(y\):\[ y = 1 - 3 = -2 \]

Key Concepts

Solving Equations

Solving equations is like finding the answer to a puzzle, where you figure out unknown values by using the information given. In math, equations are like statements that show a balance between two expressions. The key to solving these equations is to keep this balance by performing the right operations on both sides.
For example, if you are given an equation like our original one, \(y=\frac{1}{2}x - 3\), you can find the specific value of \(y\) by inserting (or substituting) the value of \(x\) provided. In this way, solving simple equations can eventually equip you with the skills needed to tackle more complex mathematical problems down the line.
To make solving easier, remember these tips:

• Isolate the variable: The variable you want to solve for should be alone on one side of the equation.
• Perform the same operation on both sides to maintain balance.
• Check your work by substituting the value back into the original equation.

In our exercise, this strategy helped us find that when \(x = 2\), the value of \(y\) becomes \(-2\).

Substitution Method

The substitution method is a powerful tool in algebra that involves replacing a variable (or variables) with their known values to simplify solving equations. It's like swapping out pieces of a puzzle that you already knew fit somewhere for those that are still missing. This method is particularly handy when you're dealing with equations involving unknown numbers.
Here's how it works: In the equation \(y=\frac{1}{2}x - 3\), we were given \(x=2\). To apply the substitution method, we simply replace the variable \(x\) with the number 2 from the equation. It's crucial to carefully perform this step as it sets the stage for the rest of the solution.
By substituting, we "translate" our theoretical equation into something concrete, allowing us to progress to a numerical answer. This is a critical step, especially for multi-step problems, as it simplifies the equation and leads directly to solving for the other variables.
As seen in our example, substituting \(x = 2\) made the equation manageable, allowing us to solve for \(y\) by further simplifying the problem. A meticulous approach to substitution helps avoid errors and ensures results are both accurate and understandable.

Prealgebra Concepts

The foundations of algebra are built on prealgebra concepts. Understanding these basics is important before moving forward into more advanced topics. Prealgebra serves as the introductory language of algebra, ensuring that students grasp the fundamental operations and properties of numbers.
Key elements of prealgebra include:

• Understanding integers and rational numbers.
• Grasping the concept of variables and how they work in equations.
• Performing basic operations—addition, subtraction, multiplication, division—with variables and numbers.
• Applying these operations to simple algebraic expressions.

In our exercise, these prealgebra concepts are at play. We simplified the expression by multiplying and subtracting, which are prealgebra operations. Such exercises help reinforce the logic and process of manipulating numbers and variables, making you more comfortable with algebra's more complex methods as you progress.
This exploration of prealgebra concepts is pivotal for building confidence and competence in handling equations of increasing difficulty.