Find the Slope and y-intercept y+1/2x=0
The question is asking for two pieces of information about the linear equation provided. The equation given is y + (1/2)x = 0, and the task is to:
Find the slope: The slope of a line in a two-dimensional space represents the rate of change of the y-coordinate with respect to the x-coordinate. It's a measure of how steep the line is and is often denoted as "m" in the slope-intercept form of a line, which is y = mx + b.
Find the y-intercept: The y-intercept is the point at which the line crosses the y-axis on a graph. At this point, the value of x is zero. It is represented as "b" in the slope-intercept form.
$y + \frac{1}{2} x = 0$
Answer
Solution:
Step:1
Transform the equation into the slope-intercept form.
Step:1.1
The standard form of the slope-intercept equation is $y = mx + b$, where $m$ represents the slope and $b$ is the y-intercept.
Step:1.2
Organize the terms on the left-hand side.
Step:1.2.1
Combine the terms $\frac{1}{2}$ and $x$ to get $y + \frac{x}{2} = 0$.
Step:1.3
Isolate $y$ by subtracting $\frac{x}{2}$ from both sides, resulting in $y = -\frac{x}{2}$.
Step:1.4
Express the equation in the form of $y = mx + b$.
Step:1.4.1
Rearrange the terms to align with the slope-intercept form, yielding $y = -\left(\frac{1}{2}x\right)$.
Step:1.4.2
Eliminate the parentheses to simplify the equation to $y = -\frac{1}{2}x$.
Step:2
Identify the slope and y-intercept from the slope-intercept form.
Step:2.1
Determine the values of $m$ (slope) and $b$ (y-intercept) from the equation $y = mx + b$. Here, $m = -\frac{1}{2}$ and $b = 0$.
Step:2.2
The slope of the line is given by the coefficient of $x$, which is $m$, and the y-intercept is the constant term $b$. Therefore, the slope is $-\frac{1}{2}$ and the y-intercept is at the point $(0, 0)$.
Knowledge Notes:
The problem involves finding the slope and y-intercept of a linear equation. The slope-intercept form of a linear equation is a convenient way to directly read off the slope and y-intercept of the line it represents. The general form of this equation is $y = mx + b$, where:
$m$ is the slope of the line, indicating how steep the line is. It is the ratio of the change in $y$ to the change in $x$ (rise over run).
$b$ is the y-intercept, which is the point where the line crosses the y-axis. This occurs when $x = 0$.
To convert a linear equation to the slope-intercept form, one must solve the equation for $y$ in terms of $x$ and constants. Once the equation is in this form, the slope and y-intercept can be directly identified as the coefficients corresponding to $x$ and the constant term, respectively.
In the given problem, the equation $y + \frac{1}{2}x = 0$ is already close to the slope-intercept form but needs to be rearranged to isolate $y$. By moving the term with $x$ to the other side, we get $y = -\frac{1}{2}x$, which is now in the slope-intercept form with $m = -\frac{1}{2}$ and $b = 0$. This indicates that the slope of the line is $-\frac{1}{2}$, and it crosses the y-axis at the origin $(0, 0)$.
