Tara is graphing the equation [tex]$4 x+2 y=10$[/tex]. Which of these shows the correct equation in slope-intercept form, slope, and [tex]$y$[/tex]-intercept?
[tex]$y=-2 x+5$[/tex]
slope [tex]$=-2$[/tex]
[tex]$y$[/tex]-intercept [tex]$=5$[/tex]
[tex]$y=-4 x+5$[/tex]
slope [tex]$=-4$[/tex]
[tex]$y$[/tex]-intercept [tex]$=5$[/tex]
[tex]$y=2 x+4$[/tex]
slope [tex]$=2$[/tex]
[tex]$y$[/tex]-intercept [tex]$=4$[/tex]
[tex]$y=4 x+5$[/tex]
slope [tex]$=4$[/tex]
[tex]$y$[/tex]-intercept [tex]$=5$[/tex]
Answer (1)
Convert the equation to slope-intercept form: y = − 2 x + 5 .
Identify the slope: m = − 2 .
Identify the y-intercept: b = 5 .
The correct equation, slope, and y-intercept are y = − 2 x + 5 , slope = − 2 , and y -intercept = 5 . y = − 2 x + 5 ; s l o p e = − 2 ; y − in t erce pt = 5
Explanation
Understanding the Problem We are given the equation 4 x + 2 y = 10 and asked to find its slope-intercept form, slope, and y-intercept. The slope-intercept form of a linear equation is y = m x + b , where m is the slope and b is the y-intercept.
Isolating y To convert the given equation to slope-intercept form, we need to isolate y on one side of the equation. First, subtract 4 x from both sides:
4 x + 2 y − 4 x = 10 − 4 x
2 y = − 4 x + 10
Slope-intercept form Next, divide both sides of the equation by 2:
2 2 y = 2 − 4 x + 10
y = − 2 x + 5
Identifying Slope and y-intercept Now that the equation is in slope-intercept form, we can identify the slope and y-intercept. The slope is the coefficient of x , which is − 2 . The y-intercept is the constant term, which is 5 .
Final Answer Therefore, the correct equation in slope-intercept form is y = − 2 x + 5 , the slope is − 2 , and the y-intercept is 5 .
Examples
Understanding slope-intercept form is crucial in many real-world applications. For instance, imagine you're tracking the cost of a taxi ride. The initial fee is $5 (y-intercept), and you pay $2 for every mile traveled (slope). The equation y = 2 x + 5 models the total cost (y) based on the number of miles (x). This concept extends to budgeting, where you can predict expenses based on fixed costs and variable rates, or in physics, where you can describe the motion of an object with constant velocity.
