Problem 14 - Algebra 1

Question

Tell which equation you would use to isolate a variable. Explain your reasoning. $$\begin{aligned} &2 x+y=-10\\\ &3 x-y=0 \end{aligned}$$

Step-by-Step Solution

Answer

The second equation \(3x - y = 0\) is more suitable for isolating a variable because it doesn't contain any constant terms unlike the first equation.

Step 1: Define the Problem

In this problem, we aim to isolate a variable. Isolating a variable means rearranging the equation in such a way that we end up with 'Variable = ...' on one side of the equation. It simplifies further operations, especially in systems of linear equations.

Step 2: Analyze Given Equations

First equation: \(2x + y = -10\). Second equation: \(3x - y = 0\). Here, we can see that the second equation doesn't have constant terms on the left side or right side which could simplify the isolation of either \(x\) or \(y\). The first equation has a negative constant which might require additional steps if we try to isolate a variable there.

Step 3: Make the final decision

Considering the nature of these two equations, isolating a variable would be simpler using the second equation \(3x - y = 0\), as we can easily isolate \(y\) or \(x\) with minimal steps.

Key Concepts

Isolating Variables

Isolating a variable in an equation is like shining a spotlight on it, highlighting its value when the rest of the equation is on the other side. It involves rearranging the equation, so one variable is alone on one side. This makes understanding and solving the equation much easier.

When you start with a system of linear equations, choose an equation that makes it simple to put a variable by itself. In our example, the equation \(3x - y = 0\) is a better pick for isolating than \(2x + y = -10\), because it doesn't involve dealing with additional constant terms.

• Choose the equation with fewer steps needed to isolate a variable.
• Look for equations where one term might easily be moved across the equals sign.
• Simplifying isolation first can save time and reduce mistakes later on.

By deciding on the right equation to isolate first, you sketch out an easier path to solve the whole system.

Linear Equations

Linear equations are equations of the first degree, meaning they involve only sums and products of variables raised to the power of one. They form straight lines when graphed and have a limited, consistent increase or decrease rate with no curves.

The system of linear equations in our example consists of \(2x + y = -10\) and \(3x - y = 0\). These are called 'linear' because:

• Each term is a constant or the product of a constant and a single variable.
• They graph as straight lines on a coordinate plane.
• Solutions to these equations occur where the lines intersect.

Linear equations often serve as basics in algebra because they lay the groundwork for understanding more complex mathematical structures. Recognizing their properties can greatly aid in solving and manipulating equations.

Solving Equations

Solving equations is the process of determining unknown values that make the equation true. It's like finding the missing piece of a puzzle that completes the picture. In the context of linear equations, solving often involves working simultaneously with two or more equations.

To solve our given system, once a variable is isolated, you can use various methods:

• Substitution: Replace the isolated variable in the other equation(s) and solve.
• Elimination: Add or subtract equations to cancel out a variable and solve for the other.
• Graphical method: Graph the equations and find their intersection, the point(s) where they meet.

The strategy you pick can depend on the system's complexity and personal preference. With practice, solving equations becomes intuitive, providing clarity and precision in tackling mathematical challenges.