Problem 15 - Algebra 1: Concepts and Skills

Question

Tell which equation you would use to isolate a variable. Explain. $$ \begin{aligned} 3 x+5 y &=25 \\ x-2 y &=-10 \end{aligned} $$

Step-by-Step Solution

Answer

Equation 2: \(x - 2y = -10\) is the best to isolate a variable due to the 'x' having a coefficient of 1 and already standing alone on one side of the equation, which makes isolating 'x' in terms of 'y' straightforward.

Step 1: Analyze Both Equations

Equation 1 is \(3x + 5y = 25\) and Equation 2 is \(x - 2y = -10\). Both equations contain the variables 'x' and 'y', but the difference between them lies in the coefficients of the variables. In Equation 1, 'x' and 'y' are both multiplied by a number other than 1, making it a bit more complicated to isolate a variable. In Equation 2, however, 'x' has a coefficient of 1 and 'y' has a coefficient of -2.

Step 2: Choose the Easier Equation

From an observation in Step 1, it is understood that Equation 2, \(x - 2y = -10\), will be an easier one to isolate a variable from, specifically the 'x'. This is because 'x' in Equation 2 is already isolated on one side.

Step 3: Justify the Choice

We chose equation 2, \(x - 2y = -10\), because the 'x' variable already stands alone on one side of the equation. This means it would take fewer algebraic steps to isolate 'x' in terms of 'y' compared to doing the same in equation 1, thus making the simplification process smoother and faster.

Key Concepts

Understanding Linear Equations

Linear equations are mathematical expressions that represent a straight line when plotted on a graph. Each equation is made up of variables and constants.
For example, in the equation \( 3x + 5y = 25 \), 'x' and 'y' are the variables. These are the unknowns that we seek to solve. The numbers 3 and 5 are coefficients, and 25 is the constant term.
Linear equations like these form the foundation of algebraic problem-solving, allowing us to model real-world situations through mathematical expressions.

• Linear equations typically involve two variables and appear in the form \( ax + by = c \), where 'a', 'b', and 'c' are constants.
• The goal is often to find the values of 'x' and 'y' that make the equation true.

Solving linear equations involves steps such as isolating variables and simplifying terms, which are crucial concepts to master in algebra.

The Art of Variable Isolation

Variable isolation is a critical step in solving linear equations. The purpose is to get one variable on one side of the equation so that it can be solved. In many cases, this starts with identifying an equation where one variable is easier to separate.
From the exercise, we chose the equation \( x - 2y = -10 \). Here, 'x' can be isolated more easily because it already appears on one side. To isolate 'x', follow these steps:

• Add 2y to both sides of the equation to cancel out the '-2y' on the left side. That results in \( x = 2y - 10 \).
• Now, you have isolated 'x' in terms of 'y', meaning you can express x using y.

This process simplifies the equation and allows you to solve for one variable at a time. The key is understanding which equation gives you the quickest path to isolation.

Mastering Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form. This makes equations easier to work with and solutions more apparent.
Once you have isolated a variable, like 'x', in the equation \( x = 2y - 10 \), you can use this simplified form to substitute into other equations.

• Substituting this form into another equation helps to solve for 'y' once 'x' is defined.
• After finding the value of 'y', it becomes straightforward to solve for 'x' using its isolated expression.

Simplifying each step not only saves time but reduces the risk of errors. It is important to carefully handle both addition and multiplication to maintain the balance of equations. Consistently applying simplification keeps algebra manageable and efficient.