Math Problem Statement

Find the equation of the linear function whose graph is given here. Use the letter x in your answer rather than the multiplication symbol.

Solution

To find the equation of the linear function whose graph is shown, we need to determine two key elements:

  1. Slope (m): This measures the steepness and direction of the line.
  2. Y-intercept (b): This is the point where the line crosses the y-axis.

Step 1: Identify Two Points on the Line

From the graph, we can identify two points:

  • Point A: (2, 7)
  • Point B: (-3, -2)

Step 2: Calculate the Slope (m)

The slope formula is: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1} Using the points (2, 7) and (-3, -2): m=2732=95=95m = \frac{-2 - 7}{-3 - 2} = \frac{-9}{-5} = \frac{9}{5} The slope m=95m = \frac{9}{5}.

Step 3: Find the Y-intercept (b)

We use the slope-intercept form of the equation: y=mx+by = mx + b Substitute one of the points into the equation. Using point (2, 7): 7=95(2)+b7 = \frac{9}{5}(2) + b 7=185+b7 = \frac{18}{5} + b Subtract 185\frac{18}{5} from both sides to solve for bb: b=7185=355185=175b = 7 - \frac{18}{5} = \frac{35}{5} - \frac{18}{5} = \frac{17}{5}

Final Equation

The equation of the line is: y=95x+175y = \frac{9}{5}x + \frac{17}{5}

Would you like further details or have any questions?

Here are five related questions to expand your understanding:

  1. How would the equation change if the slope was negative?
  2. What does the y-intercept tell you about the graph of the line?
  3. Can you find the x-intercept for this equation?
  4. How would you write the equation in standard form Ax+By=CAx + By = C?
  5. How do you find the slope if you know the angle of the line?

Tip: Always check the signs of your slope and y-intercept carefully; they indicate the direction and position of the line on the graph.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope-Intercept Form

Formulas

Slope Formula m = (y2 - y1) / (x2 - x1)
Slope-Intercept Form y = mx + b

Theorems

Linear Equation Theorem

Suitable Grade Level

Grades 8-10