-x - 2y = -1 \quad \text(3) - High Altitude Science

Understanding the Equation –x – 2y = –1 (3): A Comprehensive Guide for Students and Learners

When it comes to linear equations in two variables, mastery of rearranged and simplified forms is essential. One such equation tutored in algebraic problem-solving is:

–x – 2y = –1 (3)

Understanding the Context

Whether you’re a high school student, a homeschool learner, or a professional sharpening foundational math skills, understanding this equation helps build confidence in algebra. This article breaks down the equation, explores its equivalent forms, and explains practical applications.

What Is the Equation –x – 2y = –1 (3)?

The expression –x – 2y = –1 is a linear equation with two variables, x and y. It represents a straight line on the Cartesian coordinate plane when graphed. Though this form is not always simplified, it correctly expresses the linear relationship between x and y.

$-x - 2y = -1 \quad \text{(3)}$

Key Insights

Step 1: Rewriting for Clarity

Although the equation —x – 2y = –1 looks complete, expressing it clearly helps in solving for variables or graphing:

Rewriting:
x + 2y = 1 (by multiplying both sides by –1)
This equivalent form (x + 2y = 1) is often easier to interpret and solve in practice.

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Final Thoughts

Step 2: Solving for One Variable

To isolate x or y—the most common algebraic manipulation—choose one variable. For instance:

Solve for x:
Start with —x – 2y = –1
Add 2y to both sides: —x = 2y – 1
Multiply both sides by –1: x = –2y + 1

Solve for y:
Start with —x – 2y = –1
Add x to both sides: –2y = x – 1
Divide both sides by –2: y = –½x + ½

These linear transformations let you express one variable in terms of the other—useful in graphing or subsitution.

Step 3: Graphing the Line

The equation x + 2y = 1 reveals critical graph features:

Slope: Rewrite as y = (–1/2)x + ½ → slope = –½
Y-intercept: (0, ½)
X-intercept: When y = 0, x = 1 → (1, 0)

Plot these points and draw a straight line. This visual representation confirms the equation models a linear relationship.