Althea

High School Teacher - Tutor for 3 years

MARCH 16, 2024

The intersection of two lines refers to a point where two distinct lines in a plane meet or cross each other. This concept is fundamental in geometry and linear algebra. Each line can be represented by an equation, typically in the form $y = mx + b$ for lines in a two-dimensional plane, where $m$ is the slope of the line and $b$ is the y-intercept.

To find the intersection of two lines, their equations are set equal to each other, and this system of equations is solved to find the values of $x$ and $y$ that satisfy both equations simultaneously. The solution $(x, y)$ represents the coordinates of the intersection point.

**Unique Intersection**: If the lines have different slopes, they will intersect at exactly one point.**No Intersection (Parallel Lines)**: If the lines have the same slope but different y-intercepts, they are parallel and do not intersect.**Infinite Intersections (Coincident Lines)**: If the lines have the same slope and the same y-intercept, they coincide with each other, effectively being the same line, and thus have an infinite number of intersections.

To find the intersection of two lines, you typically follow these steps:

**Write down the equations of the two lines.**Generally, line equations are given in the form $y = mx + b$ or $Ax + By = C$, where $m$ is the slope and $b$ is the y-intercept.**Set the right-hand sides of the equations equal to each other.**This is because at the point of intersection, both lines have the same $x$ and $y$ coordinates.**Solve the resulting equation for $x$**to find the $x$-coordinate of the intersection point.**Substitute the $x$-coordinate back into one of the original line equations**to solve for $y$, giving you the $y$-coordinate of the intersection point.

**Given Lines:**

- $3x - 2y = 6$
- $x + y = 4$

**Step 1:**Solve the second equation for $y$.

$y = 4 - x$

**Step 2: Replace $y$ in the first equation with $4 - x$ and solve for $x$.**

$3x - 2(4 - x) = 6 \Rightarrow 3x - 8 + 2x = 6 \Rightarrow 5x = 14 \Rightarrow x = \frac{14}{5}$

**Step 3: Substitute $x = \frac{14}{5}$ back into the equation for $y$.**

$y = 4 - \frac{14}{5} = \frac{20 - 14}{5} = \frac{6}{5}$

**Intersection Point:** $\left(\frac{14}{5}, \frac{6}{5}\right)$

Find the intersection of two lines by graphing the function.

$3x - 2y = 6$

$x + y = 4$

**Intersection Point:** $\left(\frac{14}{5}, \frac{6}{5}\right)$

**Given Lines:**

- $y = 2x + 1$
- $y = -x + 5$

**Step 1: Set the equations equal to each other to find $x$.** $2x + 1 = -x + 5$

**Step 2: Solve for $x$.** $3x = 4 \Rightarrow x = \frac{4}{3}$

**Step 3:** Substitute $x$ back into one of the original equations.

$y = 2(\frac{4}{3}) + 1 = \frac{8}{3} + 1 = \frac{11}{3}$

**Intersection Point:** $\left(\frac{4}{3}, \frac{11}{3}\right)$

Find the intersection of two lines by graphing the function.

$y = 2x + 1$

$y = -x + 5$

**Intersection Point:** $\left(\frac{4}{3}, \frac{11}{3}\right)$

**Given Lines:**

- $3x - 6y = 9$
- $3x - 6y = 2$

**Step 1: Rearrange Equations to Slope-Intercept Form ($y = mx + b$).**
For Line 1: $3x - 9 = 6y \Rightarrow y = \frac{1}{2}x - \frac{3}{2}$
For Line 2: $3x - 2 = 6y \Rightarrow y = \frac{1}{2}x - \frac{1}{3}$

**Step 2: Compare Slopes and Intercepts.**
Both lines have a slope of $\frac{1}{2}$, but different y-intercepts, showing they are parallel.

**Conclusion:**
No intersection.

Find the intersection of two lines by graphing the function.

$3x - 6y = 9$

$3x - 6y = 2$

**Intersection Point:** Parallel Lines, No intersection.

**Given Lines:**

- $y - 2x = 5$
- $-2y + 4x = -10$

**Step 1: Manipulate the second equation to match the form of the first.**
$-2y + 4x = -10 \Rightarrow y - 2x = 5$

**Step 2: Compare the equations.**
Both equations simplify to $y - 2x = 5$, which means they are the same line represented in identical forms.

**Conclusion:**
These are coincident lines with an infinite number of points of intersection.

Find the intersection of two lines by graphing the function.

$y - 2x = 5$

$-2y + 4x = -10$

**Intersection Point:** Coincident lines, an infinite number of points of intersection.