Parallel Lines 1

Topics

Algebra 2

Linear Equations and Graphs

Parallel Lines

Parallel Lines 1

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Question 1 of 8

1. Question
Are the two lines parallel?
- 1. Yes
- 2. No
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Parallel lines have the same gradients.

Gradient Intercept Form: $y = m x + b$ $y=mx+b$

$m$ $m$ is the gradient of the line

$b$ $b$ is the y-intercept (where the line cuts the y-axis)

The gradient is given by the coefficient of $x$ $x$ or the value of $m$ $m$ .

$y$ $y$ $=$ $=$ $2 x + 1$ $2x+1$

$m$ $m$ $=$ $=$ $2$ $2$

$y$ $y$ $=$ $=$ $2 x - 3$ $2x-3$

$m$ $m$ $=$ $=$ $2$ $2$

The two gradients are equal, so the lines are parallel.

The lines are parallel.
Question 2 of 8

2. Question
Which equations can be graphed into parallel lines?

$A. y = - 3 x + 2$ $\text(A.) y=-3x+2$

$B. y = 4 - 3 x$ $\text(B.) y=4-3x$

$C. y = - 3 + 3 x$ $\text(C.) y=-3+3x$

$D. 2 y = - 6 x + 8$ $\text(D.) 2y=-6x+8$
- 1. $B$ $B$
- 2. $A$ $A$
- 3. $D$ $D$
- 4. $C$ $C$
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Parallel lines have equal gradients.

First, list down the gradients of each equation.

Note that in the gradient-intercept form $(y =$ $(y=$ $m$ $m$ $x + b)$ $x+b)$ , $m$ $m$ is the gradient. For D, divide the equation by $2$ $2$ .

$A.$ $\text(A.)$ $y$ $y$ $=$ $=$ $- 3$ $-3$ $x + 2$ $x+2$

$m_{A}$ $m_A$ $=$ $=$ $- 3$ $-3$

$B.$ $\text(B.)$ $y$ $y$ $=$ $=$ $4$ $4$ $- 3$ $-3$ $x$ $x$

$m_{B}$ $m_B$ $=$ $=$ $- 3$ $-3$

$C.$ $\text(C.)$ $y$ $y$ $=$ $=$ $3$ $3$ $x - 3$ $x-3$

$m_{C}$ $m_C$ $=$ $=$ $3$ $3$

$D.$ $\text(D.)$ $y$ $y$ $=$ $=$ $- 3$ $-3$ $x + 4$ $x+4$

$m_{D}$ $m_D$ $=$ $=$ $- 3$ $-3$

Identify which equations have equal gradients.

$m_{A}$ $m_A$ $=$ $=$ $m_{B}$ $m_B$ $=$ $=$ $m_{D}$ $m_D$ $=$ $=$ $- 3$ $-3$

Therefore, equations $A$ $A$ , $B$ $B$ and $D$ $D$ are parallel.

$A. y = - 3 x + 2$ $\text(A.) y=-3x+2$

$B. y = 4 - 3 x$ $\text(B.) y=4-3x$

$D. 2 y = - 6 x + 8$ $\text(D.) 2y=-6x+8$

Question 3 of 8

Find the equation of the line parallel to

y = 5 - 3 x

$y=5-3x$ and passing through

(4, - 1)

$(4,-1)$

1. $y = 11 x + 3$ $y=11x+3$
2. $y = 3 x - 11$ $y=3x-11$
3. $y = 3 x + 13$ $y=3x+13$
4. $y = - 3 x + 11$ $y=-3x+11$

Question 4 of 8

Find the equation of the line parallel to $y = - 2 x + 2$ $y=-2x+2$ and passing through $(- 1, 4)$ $(-1,4)$

1. $y = 2 x + 1$ $y=2x+1$
2. $y = - 2 x + 2$ $y=-2x+2$
3. $y = 2 x + 2$ $y=2x+2$
4. $y = 2 x - 2$ $y=2x-2$

Question 5 of 8

Find the equation of the line parallel to $y=-4x+2$ and passing through $(0,4)$

1. $y=-4x-4$
2. $y=-4x+4$
3. $y=-4x-4$
4. $y=4x+4$

Question 6 of 8

Find the equation of the line parallel to $y=-3x+2$ and passing through $(-3,2)$

1. $y=-3x-7$
2. $y=3x-7$
3. $y=-3x+4$
4. $y=3x+5$

Question 7 of 8

Find the equation of the line parallel to $y=5/4x+2$ and passing through $(0,-1)$

1. $y=5/4x-1$
2. $y=5x-4$
3. $y=4x-5$
4. $y=-5x-9$

Question 8 of 8

Find the equation of the line parallel to $y=6/5x-1$ and passing through $(-5,-1)$

1. $y=6/5x+5$
2. $y=6x+25$
3. $y=-6x+5$
4. $y=5x+5/6$

$y -$ $y -$ $y_{1}$ $y_1$	$=$ $=$	$m$ $m$ $(x -$ $(x-$ $x_{1}$ $x_1$ $)$ $)$
$y -$ $y -$ $(- 1)$ $(-1)$	$=$ $=$	$(- 3)$ $(-3)$ $(x -$ $(x-$ $4$ $4$ $)$ $)$
$y + 1$ $y+1$	$=$ $=$	$- 3 (x - 4)$ $-3(x-4)$	Simplify
$y + 1$ $y+1$	$=$ $=$	$- 3 x + 12$ $-3x+12$	Distribute inside parenthesis
$y + 1$ $y+1$ $- 1$ $-1$	$=$ $=$	$- 3 x + 12$ $-3x+12$ $- 1$ $-1$	Subtract $1$ $1$ from both sides
$y$ $y$	$=$ $=$	$- 3 x + 11$ $-3x+11$

$y – color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$
$y – color(royalblue)((4))$	$=$	$color(tomato)((-2))(x- color(royalblue)(-1))$
$y-4$	$=$	$-2(x+1)$	Simplify
$y-4$	$=$	$-2x-2$	Distribute inside parenthesis
$y-4 color(crimson)(+4)$	$=$	$-2x-2 color(crimson)(+4)$	Add $4$ from both sides
$y$	$=$	$-2x+2$

$y – color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$
$y – color(royalblue)((4))$	$=$	$color(tomato)((-4))(x- color(royalblue)(0))$
$y-4$	$=$	$-4(x-0)$	Simplify
$y-4$	$=$	$-4x+0$	Distribute inside parenthesis
$y-4 color(crimson)(+4)$	$=$	$-4x color(crimson)(+4)$	Add $4$ from both sides
$y$	$=$	$-4x+4$

$y – color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$
$y – color(royalblue)((2))$	$=$	$color(tomato)((-3))(x- color(royalblue)(-3))$
$y-2$	$=$	$-3(x+3)$	Simplify
$y-2$	$=$	$-3x-9$	Distribute inside parenthesis
$y-2 color(crimson)(+2)$	$=$	$-3x-9 color(crimson)(+2)$	Add $2$ from both sides
$y$	$=$	$-3x-7$

$y – color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$
$y – color(royalblue)((-1))$	$=$	$color(tomato)((5/4))(x- color(royalblue)(0))$
$y+1$	$=$	$5/4(x-0)$	Simplify
$y+1$	$=$	$5/4x$	Distribute inside parenthesis
$y+1 color(crimson)(-1)$	$=$	$5/4x color(crimson)(-1)$	Subtract $1$ from both sides
$y$	$=$	$5/4x-1$

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Quiz summary

Information

1. Question

Hint

Great Work!

Gradient Intercept Form: $y = m x + b$ $y=mx+b$

2. Question

Hint

Fantastic!

3. Question

Hint

Great Work!

Point-Gradient Formula: $y -$ $y -$ $y_{1}$ $y_1$ $=$ $=$ $m$ $m$ $(x -$ $(x-$ $x_{1}$ $x_1$ $)$ $)$

Remember

4. Question

Keep Going!

Point Slope Form: $y - y_{1} = m (x - x_{1})$ $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

5. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

6. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

7. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

8. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Quizzes

$A.$ $\text(A.)$	$y$ $y$	$=$ $=$	$- 3$ $-3$ $x + 2$ $x+2$
	$m_{A}$ $m_A$	$=$ $=$	$- 3$ $-3$

$B.$ $\text(B.)$	$y$ $y$	$=$ $=$	$4$ $4$ $- 3$ $-3$ $x$ $x$
	$m_{B}$ $m_B$	$=$ $=$	$- 3$ $-3$

$C.$ $\text(C.)$	$y$ $y$	$=$ $=$	$3$ $3$ $x - 3$ $x-3$
	$m_{C}$ $m_C$	$=$ $=$	$3$ $3$

$D.$ $\text(D.)$	$y$ $y$	$=$ $=$	$- 3$ $-3$ $x + 4$ $x+4$
	$m_{D}$ $m_D$	$=$ $=$	$- 3$ $-3$

$y$ $y$	$=$ $=$	$5 - 3 x$ $5-3x$
$y$ $y$	$=$ $=$	$- 3$ $-3$ $x + 5$ $x+5$
$m$ $m$	$=$ $=$	$- 3$ $-3$

$y$ $y$	$=$ $=$	$- 2 x + 2$ $color(tomato)(-2)x+2$
$m$ $m$	$=$ $=$	$- 2$ $-2$

Parallel Lines 1

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Quiz summary

Information

1. Question

Hint

Great Work!

Gradient Intercept Form: y=mx+by=mx+by=mx+b

2. Question

Hint

Fantastic!

3. Question

Hint

Great Work!

Point-Gradient Formula: y-y−y -y1y1y_1== =mmm(x-(x−(x-x1x1x_1)))

Remember

4. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y−y1=m(x−x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

5. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

6. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

7. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

8. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

Quizzes

Gradient Intercept Form: $y = m x + b$ $y=mx+b$

Point-Gradient Formula: $y -$ $y -$ $y_{1}$ $y_1$ $=$ $=$ $m$ $m$ $(x -$ $(x-$ $x_{1}$ $x_1$ $)$ $)$

Point Slope Form: $y - y_{1} = m (x - x_{1})$ $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$