Math Problem Statement

Find the gradient of the line joining (3,-2) and (5, 3). What is the equation of this line?

Solution

Step 1: Find the Gradient of the Line

The formula to calculate the gradient (or slope) mm of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is:

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

For the points (3,2)(3, -2) and (5,3)(5, 3), let:

  • (x1,y1)=(3,2)(x_1, y_1) = (3, -2)
  • (x2,y2)=(5,3)(x_2, y_2) = (5, 3)

Now, applying the formula:

m=3(2)53=3+22=52m = \frac{3 - (-2)}{5 - 3} = \frac{3 + 2}{2} = \frac{5}{2}

So, the gradient of the line is 52\frac{5}{2}.


Step 2: Find the Equation of the Line

The equation of a line in slope-intercept form is:

y=mx+cy = mx + c

Where:

  • mm is the gradient
  • cc is the y-intercept

We already know that the gradient m=52m = \frac{5}{2}. To find cc, we can substitute one of the points into the equation.

Using the point (3,2)(3, -2):

2=52×3+c-2 = \frac{5}{2} \times 3 + c 2=152+c-2 = \frac{15}{2} + c c=2152c = -2 - \frac{15}{2} c=42152c = -\frac{4}{2} - \frac{15}{2} c=192c = -\frac{19}{2}

Thus, the equation of the line is:

y=52x192y = \frac{5}{2}x - \frac{19}{2}

Final Answer:

  • The gradient of the line is 52\frac{5}{2}.
  • The equation of the line is y=52x192y = \frac{5}{2}x - \frac{19}{2}.

Would you like any further details or have any questions? Here are some related questions:

  1. How do you interpret the meaning of the gradient in this context?
  2. What would the equation of the line look like in standard form?
  3. How can you find the x-intercept of this line?
  4. How would you graph this line on the coordinate plane?
  5. What happens to the equation of the line if the points were switched?

Tip: When calculating the gradient of a line, always make sure to subtract the y-coordinates and x-coordinates in the correct order to avoid sign errors.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope
Coordinate Geometry

Formulas

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

Theorems

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Suitable Grade Level

Grades 7-9