Suppose that L1 and L2 are lines in the plane, that the x-intercepts of L1 and L2 are 5
and −1, respectively, and that the respective y-intercepts are 5 and 1. Then L1 and L2
intersect at the point ( _ , _ ) .

To find the point of intersection of the two lines \( L_1 \) and \( L_2 \), we first need the equations of these lines. Since the problem provides both the x- and y-intercepts for each line, we can use this information to write the equations.

### Step 1: Find the equation of \( L_1 \)

The x-intercept of \( L_1 \) is 5, and the y-intercept is 5. The equation of a line with these intercepts can be written using the intercept form:

\[
\frac{x}{a} + \frac{y}{b} = 1
\]

where \( a \) is the x-intercept and \( b \) is the y-intercept. For \( L_1 \), the intercepts are both 5, so the equation becomes:

\[
\frac{x}{5} + \frac{y}{5} = 1
\]

Multiplying through by 5:

\[
x + y = 5
\]

### Step 2: Find the equation of \( L_2 \)

The x-intercept of \( L_2 \) is −1, and the y-intercept is 1. Again, using the intercept form:

\[
\frac{x}{a} + \frac{y}{b} = 1
\]

For \( L_2 \), the x-intercept is −1 and the y-intercept is 1, so the equation becomes:

\[
\frac{x}{-1} + \frac{y}{1} = 1
\]

Simplifying this:

\[
-x + y = 1
\]

or equivalently:

\[
x - y = -1
\]

### Step 3: Solve the system of equations

Now, we have the system of equations:

1. \( x + y = 5 \)
2. \( x - y = -1 \)

Add these two equations together:

\[
(x + y) + (x - y) = 5 + (-1)
\]

\[
2x = 4
\]

\[
x = 2
\]

Now substitute \( x = 2 \) into one of the original equations, say \( x + y = 5 \):

\[
2 + y = 5
\]

\[
y = 3
\]

### Final Answer:
The two lines intersect at the point \( (2, 3) \).

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How can you find the slope of a line if you are only given its intercepts?
2. Can you derive the point-slope form of a line from the intercept form?
3. What happens if two lines have the same slope but different intercepts?
4. How do you find the intersection point if the lines are given in slope-intercept form?
5. How can you verify if two lines are perpendicular or parallel using their equations?

**Tip:** When solving systems of linear equations, adding or subtracting the equations can simplify solving for one variable.

Suppose that L1 and L2 are lines in the plane, that the x-intercepts of L1 and L2 are 5 and −1, respectively, and that the respective y-intercepts are 5 and 1. Then L1 and L2 intersect at the point ( _ , _ ) .

Discover how to find the intersection point of two lines given their x- and y-intercepts. This problem involves linear equations and solving systems of equations, suitable for students in Grades 9-10. Learn step-by-step to solve for the intersection point (2, 3).

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