Match each linear equation with the name of its form.
| Equation | | Form |
| :---------------- | :-: | :---------------------- |
| [tex]$y+6=-3(x-1)$[/tex] | | point-slope form |
| [tex]$2 x-5 y=9$[/tex] | | standard form |
| [tex]$y=-x+8$[/tex] | | slope-intercept form |
Answer (2)
The equation y + 6 = − 3 ( x − 1 ) matches the point-slope form, 2 x − 5 y = 9 matches the standard form, and y = − x + 8 matches the slope-intercept form.
;
y + 6 = − 3 ( x − 1 ) matches with point-slope form.
2 x − 5 y = 9 matches with standard form.
y = − x + 8 matches with slope-intercept form.
The final answer is the correct matching of the equations with their corresponding forms.
Explanation
Understanding the Problem We are given three linear equations and three forms of linear equations. Our goal is to match each equation to its corresponding form.
Matching with Point-Slope Form The point-slope form of a linear equation is given by y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is a point on the line and m is the slope. The equation y + 6 = − 3 ( x − 1 ) can be rewritten as y − ( − 6 ) = − 3 ( x − 1 ) . Thus, it matches the point-slope form.
Matching with Standard Form The standard form of a linear equation is given by A x + B y = C , where A , B , and C are constants. The equation 2 x − 5 y = 9 is already in this form. Thus, it matches the standard form.
Matching with Slope-Intercept Form The slope-intercept form of a linear equation is given by y = m x + b , where m is the slope and b is the y-intercept. The equation y = − x + 8 is in this form, with m = − 1 and b = 8 . Thus, it matches the slope-intercept form.
Final Answer Therefore, the matched equations and their forms are:
y + 6 = − 3 ( x − 1 ) matches with point-slope form.
2 x − 5 y = 9 matches with standard form.
y = − x + 8 matches with slope-intercept form.
Examples
Linear equations are used in various real-life scenarios, such as calculating the cost of items, determining the distance traveled at a constant speed, or modeling the relationship between two variables. For example, if you want to determine how much it costs to take a taxi, you can use a linear equation where the cost is dependent on the distance traveled. If the taxi charges a flat fee of $5 and an additional 2 p er mi l e , t h ee q u a t i o n w o u l d b e y = 2x + 5 , w h ere y i s t h e t o t a l cos t an d x$ is the number of miles traveled. Understanding the different forms of linear equations helps in analyzing and solving such problems efficiently.
