Write the equation in slope-intercept form. [tex]y=2(5 x+1)+3(5 x+3)[/tex]
A. [tex]y=26 x+12[/tex]
B. [tex]y=25 x+12[/tex]
C. [tex]y=26 x+11[/tex]
D. [tex]y=25 x+11[/tex]
Answer (2)
To write the equation y = 2 ( 5 x + 1 ) + 3 ( 5 x + 3 ) in slope-intercept form:
Distribute the constants: y = 10 x + 2 + 15 x + 9 .
Combine like terms: y = ( 10 x + 15 x ) + ( 2 + 9 ) .
Simplify to slope-intercept form: y = 25 x + 11 .
Explanation
Understanding the Problem We are given the equation y = 2 ( 5 x + 1 ) + 3 ( 5 x + 3 ) and asked to write it in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept.
Distributing Constants First, we distribute the constants in the equation:
y = 2 ( 5 x + 1 ) + 3 ( 5 x + 3 ) becomes
y = 10 x + 2 + 15 x + 9
Combining Like Terms Next, we combine like terms:
y = ( 10 x + 15 x ) + ( 2 + 9 )
Simplifying the Equation Finally, we simplify the equation to the slope-intercept form:
y = 25 x + 11
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 like the y-intercept, and the cost per mile is the slope. If the initial fee is $11 and the cost per mile is $25, the equation y = 25x + 11 models the total cost (y) for x miles traveled. This concept extends to budgeting, where you can predict expenses based on a fixed starting amount and a consistent rate of spending, or in physics, where you can describe the motion of an object with constant velocity.
The equation y = 2 ( 5 x + 1 ) + 3 ( 5 x + 3 ) simplifies to y = 25 x + 11 in slope-intercept form. Therefore, the correct answer is option D: y = 25 x + 11 .
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