Math Problem Statement

Which equation in standard form has a graph that passes through the point (−4, 2)(−4, 2) and has a slope of 9292 ?

A.9x−2y=369x−2y=36

B.9x−2y=269x−2y=26

C.9x−2y=−409x−2y=−40

D.9x−2y=−109x−2y=−10

Solution

To determine which equation in standard form represents the line that passes through the point (4,2)(-4, 2) and has a slope of 92\frac{9}{2}, we need to use the point-slope form of the equation of a line and convert it to standard form.

  1. Point-slope form of the equation: yy1=m(xx1)y - y_1 = m(x - x_1) where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope.

  2. Substituting (x1,y1)=(4,2)(x_1, y_1) = (-4, 2) and m=92m = \frac{9}{2} into the point-slope form: y2=92(x+4)y - 2 = \frac{9}{2}(x + 4)

  3. Simplify and convert to slope-intercept form: y2=92x+924y - 2 = \frac{9}{2}x + \frac{9}{2} \cdot 4 y2=92x+18y - 2 = \frac{9}{2}x + 18 y=92x+18+2y = \frac{9}{2}x + 18 + 2 y=92x+20y = \frac{9}{2}x + 20

  4. Convert to standard form Ax+By=CAx + By = C: y=92x+20y = \frac{9}{2}x + 20 Multiply every term by 2 to eliminate the fraction: 2y=9x+402y = 9x + 40 Rearrange to standard form: 9x2y=409x - 2y = -40

Thus, the correct equation in standard form is: 9x2y=409x - 2y = -40

So, the correct answer is: C. 9x - 2y = -40

Would you like details or have any questions?

Here are 5 related questions to expand your understanding:

  1. How do you convert an equation from slope-intercept form to standard form?
  2. What is the point-slope form of a line, and when is it useful?
  3. Can you explain the process of finding the slope of a line given two points?
  4. How do you determine if a point lies on a given line?
  5. What are the differences between different forms of linear equations (slope-intercept, point-slope, standard)?

Tip: Always check your work by substituting the given point into your final equation to ensure it satisfies the equation.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Point-Slope Form
Standard Form of a Line

Formulas

Point-Slope form: y - y1 = m(x - x1)
Slope-Intercept form: y = mx + b
Standard form: Ax + By = C

Theorems

Linear Equation Conversion

Suitable Grade Level

Grades 9-11