Math Problem Statement

Write an equation of the line passing through the point (-4, -4) that is parallel to the line y = 3/2x - 1.

Solution

We are asked to write the equation of a line passing through the point (4,4)(-4, -4) and parallel to the line given by the equation y=32x1y = \frac{3}{2}x - 1.

Step 1: Identify the slope

The line y=32x1y = \frac{3}{2}x - 1 is in slope-intercept form y=mx+by = mx + b, where mm is the slope. From the equation, the slope m=32m = \frac{3}{2}.

Since the new line is parallel to the given line, it will have the same slope, i.e., m=32m = \frac{3}{2}.

Step 2: Use the point-slope form

The point-slope form of a line is: yy1=m(xx1)y - y_1 = m(x - x_1) where (x1,y1)(x_1, y_1) is the point the line passes through and mm is the slope. Substituting (4,4)(-4, -4) for (x1,y1)(x_1, y_1) and 32\frac{3}{2} for mm, we get: y(4)=32(x(4))y - (-4) = \frac{3}{2}(x - (-4)) Simplifying: y+4=32(x+4)y + 4 = \frac{3}{2}(x + 4)

Step 3: Distribute the slope

Now distribute the 32\frac{3}{2} on the right-hand side: y+4=32x+32×4y + 4 = \frac{3}{2}x + \frac{3}{2} \times 4 y+4=32x+6y + 4 = \frac{3}{2}x + 6

Step 4: Solve for yy

Subtract 4 from both sides to get the equation in slope-intercept form: y=32x+64y = \frac{3}{2}x + 6 - 4 y=32x+2y = \frac{3}{2}x + 2

Thus, the equation of the line passing through the point (4,4)(-4, -4) and parallel to y=32x1y = \frac{3}{2}x - 1 is: y=32x+2y = \frac{3}{2}x + 2

Would you like further details or have any questions about this solution?

Related questions:

  1. How would the equation change if the line were perpendicular instead of parallel?
  2. What is the y-intercept of the new line?
  3. Can you explain how to convert a line equation from point-slope form to slope-intercept form?
  4. What is the general form of a linear equation?
  5. How do we determine the equation of a line passing through two given points?

Tip:

When finding equations of parallel lines, remember that they always share the same slope but have different y-intercepts unless they are the same line.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Point-Slope Form
Slope-Intercept Form

Formulas

Slope-Intercept Form: y = mx + b
Point-Slope Form: y - y1 = m(x - x1)

Theorems

Parallel Line Theorem: Parallel lines share the same slope

Suitable Grade Level

Grades 9-10