## Math 9 Chapter 2 Lesson 4: Parallel lines and intersecting lines

## 1. Summary of theory

### 1.1. Parallel lines

Let d have the equation \(y=ax+b(a\neq0)\) and the line d’ have the equation \(y=a’x+b'(a’\neq0)\). Then d and d’ are parallel if and only if \(a=a’\) and \(b\neq b’\)

Note: if a=a’ and b=b’ then d coincides with d’

### 1.2. Straight lines intersect

Let the line d have the equation \(y=ax+b(a\neq0)\) and the line d’ have the equation \(y=a’x+b'(a’\neq0)\). Then d and d’ intersect if and only if \(a\neq a’\)

Note: if \(a\neq a’\) and b=b’ then d intersects d’ at a point on the vertical axis with coordinates b

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Find the value of m so that the two lines \(y=2x+3\) and \(y=(m-1)x+2\) are parallel.

**Solution guide**

For two given lines to be parallel, then \(2=m-1\) or \(m=3\)

**Verse 2: **Find the value of m so that the two lines \(y=mx+1\) and \(y=(3m-4)x-2\) intersect.

**Solution guide**

For two given lines to intersect, then \(m \neq 3m-4\) or \(m \neq 2\)

**Question 3: **Find the equation of the line d knowing that d is parallel to the line \(y=2x+1\) and d passes through \(A(1;2)\)

**Solution guide**

Call \((d):y=ax+b\), d is parallel to \(y=2x+1\) so \(a=2\), d goes through \(A(1;2)\) so \(2=2.1+b\) or \(b=0\), i.e. \((d):y=2x\)

### 2.2. Advanced exercises

**Question 1: **Determine the coefficients a and b so that the line \(y=ax+b\) intersects the vertical axis at a point with coordinates equal to -2 and is parallel to the line OA, where O is the origin, \(A (2;1).\)

**Solution guide**

The line OA has the equation \(x-2y=0\) so \(a=\frac{1}{2}\). On the other hand, the given line passes through the point \((0;-2)\) so we can find \(b=-2\)

**Verse 2: **Given three points \(A(-1;6), B(-4;4), C(1;1)\). Find the coordinates D such that \(ABCD\) is a parallelogram.

**Solution guide**

The line AB has the equation \(2x-3y+20=0\)

The line BC has the equation \(3x+5y-8=0\)

The line through A and parallel to BC has the equation \(3x+5y-27=0\)

The line through C and parallel to AB has the equation \(2x-3y+1=0\)

Then coordinate D is the solution of the system of equations \(\left\{\begin{matrix}3x+5y=27\\2x-3y=-1 \end{matrix}\right.\)

Solving the system we get \(\left\{\begin{matrix} x=4\\ y=3 \end{matrix}\right.\). So \(D(4;3)\)

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Indicate three pairs of intersecting lines and pairs of parallel lines among the following lines:

a) \(y = 1.5x + 2\) b) \(y = x + 2\)

c) \(y = 0.5x – 3\) d) \(y = x – 3\)

e) \(y = 1.5x – 1\) g) \(y = 0.5x + 3\)

**Verse 2: **Given the first order functions \(y = mx + 3\) and \(y = (2m + 1)x – 5\). Find the value of m so that the graph of the two given functions is:

a) Two lines are parallel to each other;

b) Two straight lines intersect.

**Question 3: **Given the function \(y = ax + 3\). Determine the coefficient a in each of the following cases:

a) The graph of the function is parallel to the line \(y = -2x\);

b) When \(x = 1 + \sqrt 2\) then \(y = 2 + \sqrt 2 \).

**Question 4:** Find the coefficient a of the function y = ax + 1

Know that when \(x = 1 + \sqrt 2\) then \(y = 3 + \sqrt 2 \)

**Question 5:** Determine the function in each of the following cases, knowing the graph of the function is a straight line passing through the origin:

a) Pass through the point \(A(3;2)\) ;

b) The coefficient a is equal to \(\sqrt 3 \) ;

c) Parallel to the line \(y = 3x + 1.\)

### 3.2. Multiple choice exercises

**Question 1: **Find the value of m so that the two lines \(y=mx+1\) and \(y=(m-4)x-2\) intersect.

A. \(m = 1\)

B. For every m

C. \(m \neq 0\)

D. Does not exist m

**Verse 2:** Determine the line d passing through the origin O and parallel to the line AB knowing that \(A(-1;1)\) and \(B(-1;3).\)

A. \(y=x\)

B. \(x+y=0\)

C. \(y=0\)

D. \(x=0\)

**Question 3: **Find the value of m so that the two lines \(y=x+3\) and \(y=(m-1)x+2\) are parallel.

A. 2

B. 1

C. -2

D. 0

**Question 4: **Given four points \(A(1;4), B(3;5), C(6;4), D(2;2)\). What is the quadrilateral \(ABCD\)?

A. Trapezoid

B. Parallelogram

C. Square trapezoid

D. Rhombus

**Question 5:** Given two lines \((d): y=ax+b\) and \((d’): y=a’x+b’\), the condition for \((d)\) and \(( d’)\) perpendicular to each other is \(aa’=-1\). Based on that find m such that the two lines \(y=2x+3\) and \(y=(m-1)x+2\) are perpendicular.

A. \(\frac{1}{2}\)

B. 2

C. 1

D. 0

## 4. Conclusion

Through this lesson, students will learn some of the main topics as follows:

- Identify and recall the condition for two lines y = ax + b (a ≠ 0) and the line y = a’x + b’ (a’ ≠ 0) to intersect, parallel, and coincide.
- Apply the theory to solve problems of finding the values of the given parameters in first-order functions so that their graphs are two intersecting, parallel, and overlapping lines.

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