Consider the system of equations:
[tex]
\begin{array}{l}
y=-2 x+4 \\
3 y+x=-3
\end{array}
[/tex]
Which statement is true of this system of equations?
A. Both equations are in slope-intercept form.
B. The first equation converted to slope-intercept form is [tex]$y+2 x=4$[/tex].
C. The second equation converted to slope-intercept form is [tex]$y=-\frac{1}{3} x-$[/tex].
D. Neither equation is in slope-intercept form.
Answer (2)
The true statement is B: The first equation converted to slope-intercept form is y + 2 x = 4 . The first equation is already in slope-intercept form, while the second has been converted into slope-intercept form successfully. Thus, statement B accurately reflects the situation of the equations.
;
The first equation y = − 2 x + 4 is already in slope-intercept form.
Convert the second equation 3 y + x = − 3 to slope-intercept form: y = − 3 1 x − 1 .
Check each statement to see which one is true.
The true statement is: The first equation converted to slope-intercept form is y + 2 x = 4 , which is a rearrangement of the original equation. T h e f i rs t e q u a t i o n co n v er t e d t o s l o p e − in t erce pt f or m i s y + 2 x = 4
Explanation
Analyze the equations We are given a system of two equations:
y = − 2 x + 4
3 y + x = − 3
We need to determine which statement is true about this system. Let's analyze each statement.
Check the first equation The slope-intercept form of a linear equation is y = m x + b , where m is the slope and b is the y-intercept.
The first equation, y = − 2 x + 4 , is already in slope-intercept form.
Convert the second equation The second equation, 3 y + x = − 3 , is not in slope-intercept form. To convert it to slope-intercept form, we need to isolate y :
3 y + x = − 3
Subtract x from both sides:
3 y = − x − 3
Divide both sides by 3:
y = 3 − x − 3
y = − 3 1 x − 1
Check the statements Now, let's check the given statements:
"Both equations are in slope-intercept form." This is false because the second equation was not initially in slope-intercept form.
"The first equation converted to slope-intercept form is y + 2 x = 4 ." The first equation is already in slope-intercept form as y = − 2 x + 4 . If we rearrange this, we get y + 2 x = 4 . So, this statement is true.
"The second equation converted to slope-intercept form is y = − 3 1 x − ." Our calculation shows the second equation in slope-intercept form is y = − 3 1 x − 1 . The given statement is missing the constant term, so it's incomplete and thus false.
"Neither equation is in slope-intercept form." This is false because the first equation is in slope-intercept form.
Final Answer Therefore, the true statement is: The first equation converted to slope-intercept form is y + 2 x = 4 .
Examples
Understanding slope-intercept form is crucial in many real-world applications. For example, if you are tracking the distance a car travels over time, the equation might be y = 60 x + 10 , where y is the distance in miles, x is the time in hours, 60 is the speed (slope), and 10 is the initial distance (y-intercept). By converting equations to slope-intercept form, you can easily determine the rate of change and initial conditions, which helps in making predictions and informed decisions.
