Find the slope and the $y$-intercept of the line.
$9 x+3 y=-6$
Write your answers in simplest form.
Answer (2)
Rewrite the given equation in slope-intercept form: y = m x + b .
Isolate the y term: 3 y = − 9 x − 6 .
Divide by 3 to get: y = − 3 x − 2 .
Identify the slope and y-intercept: slope = − 3 , y -intercept = − 2 .
Explanation
Understanding the Problem We are given the equation of a line: 9 x + 3 y = − 6 . Our goal is to find the slope and the y -intercept of this line. To do this, we will rewrite the equation in slope-intercept form, which is y = m x + b , where m represents the slope and b represents the y -intercept.
Isolating the y term First, we need to isolate the y term. We start by subtracting 9 x from both sides of the equation:
9 x + 3 y − 9 x = − 6 − 9 x
This simplifies to:
3 y = − 9 x − 6
Solving for y Next, we divide both sides of the equation by 3 to solve for y :
3 3 y = 3 − 9 x − 6
This simplifies to:
y = − 3 x − 2
Identifying Slope and y-intercept Now that the equation is in slope-intercept form ( y = m x + b ), we can easily identify the slope and the y -intercept.
The slope, m , is the coefficient of x , which is − 3 .
The y -intercept, b , is the constant term, which is − 2 .
Final Answer Therefore, the slope of the line is − 3 and the y -intercept is − 2 .
Examples
Understanding the slope and y-intercept of a line is crucial in many real-world applications. For example, if you're tracking the cost of a taxi ride, the y-intercept could represent the initial fare, and the slope could represent the cost per mile. Similarly, in physics, the slope of a velocity-time graph represents acceleration, and the y-intercept represents the initial velocity. By understanding these concepts, you can model and analyze various linear relationships in everyday situations.
The slope of the line is − 3 , and the y -intercept is − 2 . These values are derived from rewriting the line equation in slope-intercept form, y = m x + b .
;
