Math Problem Statement

Find an equation of the line through (4,5) and parallel to 8x - 2y = 1

Solution

We are asked to find the equation of a line that passes through the point (4,5)(4, 5) and is parallel to the line given by 8x2y=18x - 2y = 1.

Step 1: Determine the slope of the given line

We can first find the slope of the line 8x2y=18x - 2y = 1 by rewriting it in slope-intercept form, y=mx+by = mx + b, where mm is the slope.

8x2y=18x - 2y = 1

Solve for yy:

2y=8x+1-2y = -8x + 1

y=4x12y = 4x - \frac{1}{2}

From this equation, we can see that the slope mm of the given line is 44.

Step 2: Use the slope-point form

Since the line we need to find is parallel to the given line, it will have the same slope. Therefore, the slope of the new line is also 44.

Now, using the point-slope form of a line equation:

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

Where m=4m = 4, (x1,y1)=(4,5)(x_1, y_1) = (4, 5). Substituting these values:

y5=4(x4)y - 5 = 4(x - 4)

Step 3: Simplify the equation

y5=4x16y - 5 = 4x - 16

y=4x16+5y = 4x - 16 + 5

y=4x11y = 4x - 11

Final Answer:

The equation of the line passing through (4,5)(4, 5) and parallel to 8x2y=18x - 2y = 1 is:

y=4x11y = 4x - 11

Would you like further details or clarification?

Related Questions:

  1. How do you find the slope of a line from its equation?
  2. What is the general form of the point-slope equation of a line?
  3. How can you verify if two lines are parallel?
  4. How do you convert a linear equation from standard form to slope-intercept form?
  5. Can a line parallel to another line ever intersect it?

Tip:

When two lines are parallel, their slopes are always equal! Use this to your advantage when writing the equation of a line.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

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

Theorems

Parallel Lines Theorem

Suitable Grade Level

Grades 8-10