Problem 42 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -y=x+1 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -1, and the y-intercept is -1.

Step 1: Rewrite the equation in slope-intercept form

To rewrite \(-y = x + 1\) in the form of \(y = mx + b\), we need to isolate y on one side of the equation. We can do this by simply multiplying both sides of the equation by -1. $$ (-1)(-y) = (-1)(x + 1) $$ Thus, the equation becomes: $$ y = -x - 1 $$ Now, we have the equation in slope-intercept form.

Step 2: Identify the slope and y-intercept

With the equation in the form \(y = mx + b\), we can easily identify the slope and y-intercept: Slope (\(m\)): The coefficient of the x-term is \(-1\). Therefore, the slope is \(-1\). y-intercept (\(b\)): The constant term in the equation is \(-1\), which represents the y-intercept. In other words, the line intersects the y-axis at the point \((0,-1)\).

Step 3: Answer

The slope of the line is -1, and the y-intercept is -1.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of a linear equation is essential for graphing straight lines and analyzing their properties quickly. It is expressed as y = mx + b, where m represents the slope, or the steepness, of the line, and b indicates the y-intercept, the point where the line crosses the y-axis.

Isolating Variables

When working with equations, isolating the variable of interest, usually y, is a recurring task. It involves manipulating the equation so that we have one variable on one side and all the other terms on the opposite side. In terms of linear equations, this means getting y by itself on one side of the equals sign. This can often be done through operations such as addition, subtraction, multiplication, or division applied to both sides of the equation.

Coefficient of Linear Equation

In a linear equation, the coefficient refers to the number multiplied by the variable. For instance, in the slope-intercept form y = mx + b, the coefficient m is particularly important as it gives us the slope of the line. This coefficient tells us both the direction and the steepness of the line. If m is positive, the line rises to the right; if it's negative, it falls to the right. The greater the absolute value of m, the steeper the line's incline or decline.