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Answer :
To solve the problem of determining which statements best represent the equation [tex]\( y = -12x + 24 \)[/tex], we need to understand the components of a linear equation in slope-intercept form, which is [tex]\( y = mx + c \)[/tex]. Here, [tex]\( m \)[/tex] is the slope, and [tex]\( c \)[/tex] is the y-intercept.
Let's break it down step by step:
1. Identify the Slope:
The slope [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex] in the equation. In the equation [tex]\( y = -12x + 24 \)[/tex], the coefficient of [tex]\( x \)[/tex] is [tex]\(-12\)[/tex].
So, the slope is [tex]\(-12\)[/tex].
2. Identify the Y-intercept:
The y-intercept [tex]\( c \)[/tex] is the constant term in the equation, which is the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0.
In the equation [tex]\( y = -12x + 24 \)[/tex], the constant term is [tex]\( 24 \)[/tex].
Hence, the y-intercept is [tex]\( 24 \)[/tex].
3. Determine if the Equation Represents a Linear or Nonlinear Line:
This equation, [tex]\( y = -12x + 24 \)[/tex], is in the form [tex]\( y = mx + c \)[/tex], which is the standard form for a linear equation. An equation is linear if it can be written in this form and involves no exponents higher than 1 for the variable [tex]\( x \)[/tex]. Since both conditions are met, the line is linear.
Now, let's match these findings to the provided statements:
- B. The slope is [tex]\(-12\)[/tex]. (True)
- D. The y-intercept is [tex]\( 24 \)[/tex]. (True)
- E. The line is linear. (True)
So, the correct statements that best represent the equation are B, D, and E.
Let's break it down step by step:
1. Identify the Slope:
The slope [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex] in the equation. In the equation [tex]\( y = -12x + 24 \)[/tex], the coefficient of [tex]\( x \)[/tex] is [tex]\(-12\)[/tex].
So, the slope is [tex]\(-12\)[/tex].
2. Identify the Y-intercept:
The y-intercept [tex]\( c \)[/tex] is the constant term in the equation, which is the value of [tex]\( y \)[/tex] when [tex]\( x \)[/tex] is 0.
In the equation [tex]\( y = -12x + 24 \)[/tex], the constant term is [tex]\( 24 \)[/tex].
Hence, the y-intercept is [tex]\( 24 \)[/tex].
3. Determine if the Equation Represents a Linear or Nonlinear Line:
This equation, [tex]\( y = -12x + 24 \)[/tex], is in the form [tex]\( y = mx + c \)[/tex], which is the standard form for a linear equation. An equation is linear if it can be written in this form and involves no exponents higher than 1 for the variable [tex]\( x \)[/tex]. Since both conditions are met, the line is linear.
Now, let's match these findings to the provided statements:
- B. The slope is [tex]\(-12\)[/tex]. (True)
- D. The y-intercept is [tex]\( 24 \)[/tex]. (True)
- E. The line is linear. (True)
So, the correct statements that best represent the equation are B, D, and E.
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