Math Problem Statement

A straight line has a gradient of -1 and intercepts the y-axis at 5. A second line has the equation of y=0.5+2. where do these two lines intersect?

Solution

To find the intersection of the two lines, we need to solve the system of equations formed by their respective equations.

Step 1: Identify the equations of the two lines.

  1. First line:
    Given: The gradient (slope) is 1-1 and the y-intercept is 55.
    The equation of the first line can be written as: y=x+5y = -x + 5

  2. Second line:
    The equation of the second line is given as: y=0.5x+2y = 0.5x + 2

Step 2: Set the equations equal to find the intersection.

To find the point of intersection, set the two equations equal to each other: x+5=0.5x+2-x + 5 = 0.5x + 2

Step 3: Solve for xx.

To solve for xx, first, get all the terms involving xx on one side: 52=0.5x+x5 - 2 = 0.5x + x 3=1.5x3 = 1.5x x=31.5=2x = \frac{3}{1.5} = 2

Step 4: Find the corresponding yy value.

Now that we have x=2x = 2, substitute it back into either equation to find the corresponding yy value. Let's use the first equation: y=2+5=3y = -2 + 5 = 3

Conclusion: Intersection point

The two lines intersect at the point (2,3)(2, 3).

Would you like more details, or do you have any questions?

Here are 5 related questions:

  1. How do you determine the gradient of a line from two given points?
  2. What is the geometric interpretation of the y-intercept of a line?
  3. How can you determine if two lines are parallel?
  4. How do you find the equation of a line given its slope and a point on the line?
  5. What does it mean for two lines to be perpendicular?

Tip: When solving systems of linear equations, setting the equations equal to each other is a key step in finding the intersection.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Intersection of Lines

Formulas

Equation of a line: y = mx + c

Theorems

Slope-Intercept Form

Suitable Grade Level

Grades 8-10