Math Problem Statement

The point with coordinates (d, 4) lies on the line shown. What is the value of d?

Solution

To find the value of dd for the point (d,4)(d, 4) that lies on the line, we need to determine the equation of the line shown in the graph. Let's break this down:

1. Determine the slope (m) of the line:

We can pick two clear points from the graph to calculate the slope. The points (0,6)(0, 6) and (8,0)(8, 0) are on the line.

The slope mm of a line is calculated as: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Using the points (0,6)(0, 6) and (8,0)(8, 0): m=0680=68=34m = \frac{0 - 6}{8 - 0} = \frac{-6}{8} = -\frac{3}{4}

2. Write the equation of the line:

The general equation of a line is: y=mx+by = mx + b We know the slope m=34m = -\frac{3}{4}, and we can substitute the point (0,6)(0, 6) to find the y-intercept bb: 6=34(0)+bb=66 = -\frac{3}{4}(0) + b \quad \Rightarrow \quad b = 6 So, the equation of the line is: y=34x+6y = -\frac{3}{4}x + 6

3. Substitute the known yy-coordinate:

We know the point (d,4)(d, 4) lies on the line, so substitute y=4y = 4 into the equation: 4=34d+64 = -\frac{3}{4}d + 6

4. Solve for dd:

Subtract 6 from both sides: 46=34d4 - 6 = -\frac{3}{4}d 2=34d-2 = -\frac{3}{4}d Multiply both sides by 43-\frac{4}{3}: d=2×43=83d = -2 \times -\frac{4}{3} = \frac{8}{3}

Thus, the value of dd is 83\frac{8}{3}.

However, none of the multiple-choice options exactly match this value, which suggests a need to recheck the calculations or the graph. Would you like a deeper breakdown or explanation of why this might not align with the provided options? Here are 5 related questions for expansion:

  1. How do we determine the intercepts of a line from a graph?
  2. What is the process for finding the slope if the points are unclear on the graph?
  3. How do fractional values affect the graphing of lines?
  4. What is the significance of verifying units and axes in graph problems?
  5. Why might real-world contexts cause approximations in graph-based problems?

Tip: Always verify if the graph scale or units are uniform across axes when dealing with multiple-choice problems involving graphs!

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Coordinate Geometry
Slope-Intercept Form

Formulas

Slope Formula: m = (y2 - y1) / (x2 - x1)
Line Equation: y = mx + b

Theorems

Slope-Intercept Theorem

Suitable Grade Level

Grade 9-10