Math Problem Statement

write the equation of the line in slope intercept form that passes through the point (-3,7) and (5,3)

Solution

To find the equation of the line in slope-intercept form y=mx+by = mx + b that passes through the points (3,7)(-3, 7) and (5,3)(5, 3), follow these steps:

Step 1: Find the slope mm of the line.

The slope mm is given by the formula:

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

Substituting the given points (3,7)(-3, 7) and (5,3)(5, 3):

m=375(3)=48=12m = \frac{3 - 7}{5 - (-3)} = \frac{-4}{8} = -\frac{1}{2}

Step 2: Use the point-slope form of the equation.

Now, use the point-slope form of the line equation:

yy1=m(xx1)y - y_1 = m(x - x_1)

Using the slope m=12m = -\frac{1}{2} and one of the points, say (3,7)(-3, 7):

y7=12(x+3)y - 7 = -\frac{1}{2}(x + 3)

Step 3: Simplify to get the slope-intercept form.

Distribute and simplify the equation:

y7=12x32y - 7 = -\frac{1}{2}x - \frac{3}{2}

Add 7 to both sides to solve for yy:

y=12x32+142y = -\frac{1}{2}x - \frac{3}{2} + \frac{14}{2}

y=12x+112y = -\frac{1}{2}x + \frac{11}{2}

So, the equation of the line in slope-intercept form is:

y=12x+112\boxed{y = -\frac{1}{2}x + \frac{11}{2}}

Would you like further details or have any questions?

Here are some related questions you might find useful:

  1. How do you find the slope-intercept form given two points?
  2. How do you convert a point-slope equation to slope-intercept form?
  3. What is the significance of the slope in a linear equation?
  4. How do you determine if two lines are parallel?
  5. How can you find the y-intercept if the equation is given in standard form?

Tip: The slope-intercept form y=mx+by = mx + b is useful for quickly identifying the slope mm and the y-intercept bb of a line.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

Slope formula m = (y2 - y1) / (x2 - x1)
Slope-intercept form y = mx + b
Point-slope form y - y1 = m(x - x1)

Theorems

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Suitable Grade Level

Grades 8-10