Math Problem Statement

Work out the equation of the line shown below. Give your answer in the form y = mx + c, where m and c are integers or fractions in their simplest forms.

Solution

To find the equation of the line in the form y=mx+cy = mx + c, we need to determine the slope (mm) and the y-intercept (cc) of the line.

Step 1: Determine the slope (mm)

The slope of the line is calculated using two known points on the line. Looking at the graph, we can extract two points:

  • Point 1: (60,120)(-60, -120)
  • Point 2: (0,40)(0, -40)

The formula for slope is:

m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

Substitute the coordinates:

m=40(120)0(60)=40+12060=8060=43m = \frac{-40 - (-120)}{0 - (-60)} = \frac{-40 + 120}{60} = \frac{80}{60} = \frac{4}{3}

So, the slope m=43m = \frac{4}{3}.

Step 2: Determine the y-intercept (cc)

The y-intercept (cc) is where the line crosses the y-axis. From the graph, we can see that the line crosses the y-axis at y=40y = -40, so c=40c = -40.

Final equation of the line:

Now that we have m=43m = \frac{4}{3} and c=40c = -40, the equation of the line is:

y=43x40y = \frac{4}{3}x - 40

Would you like more details on any step or have any other questions?

Here are 5 related questions:

  1. How can you find the slope of a line if only given the graph?
  2. How does changing the slope (mm) affect the appearance of a line?
  3. What is the significance of the y-intercept in a linear equation?
  4. How can you determine if two lines are parallel by looking at their equations?
  5. What is the effect of changing the y-intercept (cc) on the graph of a line?

Tip: The slope (mm) tells us how steep the line is; a larger slope means a steeper line, while the sign of the slope indicates whether the line goes up or down as xx increases.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope and Intercept
Coordinate Geometry

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Equation of a line: y = mx + c

Theorems

Slope-Intercept Form

Suitable Grade Level

Grades 8-10