smc-4216-25-Straight lines

Linear Relationships, SM-Bank 062

Use the graph of $y=3x-10$ below to find the solution to the equation $3x-10=-1$. (2 marks)

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$x=3$

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$\text{To solve }3x-10=-1\ \text{graphically, find the point}$

$\text{of intersection of the lines }y=-1\ \text{and }y=3x-10$

$\text{i.e. }(3\ ,-1)$

$\therefore\ \text{The solution is }x=3$

Linear Relationships, SM-Bank 061

Use the graph of $y=7-2x$ below to find the solution to the equation $7-2x=-3$. (2 marks)

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$x=5$

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$\text{To solve }7-2x=-3\ \text{graphically, find the point}$

$\text{of intersection of the lines }y=-3\ \text{and }y=7-2x$

$\text{i.e. }(5\ ,-3)$

$\therefore\ \text{The solution is }x=5$

Linear Relationships, SM-Bank 060

Use the graph of $y=2x+3$ below to find the solution to the equation $2x+3=11$. (2 marks)

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$x=4$

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$\text{To solve }2x+3=11\ \text{graphically, find the point}$

$\text{of intersection of the lines }y=11\ \text{and }y=2x+3$

$\text{i.e. }(4\ ,\ 11)$

$\therefore\ \text{The solution is }x=4$

Linear Relationships, SM-Bank 059

Verify that the points $(1\ ,\ -1)$ and $(-7 ,\ 3)$ lie on the line $x+2y=-1$? (3 marks)

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$\text{See worked solution}$

Show Worked Solution

$\text{Check points by substituting into }x+2y=-1$

$(1\ ,-1) \longrightarrow$	$LHS$	$=1+2\times (-1)$
		$=1-2=-1$
		$=RHS$

$(-7 ,\ 3) \longrightarrow$	$LHS$	$=-7+2\times 3$
		$=-7+6=-1$
		$=RHS$

$\therefore\ (1\ ,-1)\ \text{and }(-7 ,\ 3) \text{ both lie on the line}\ \ x+2y=-1$

Linear Relationships, SM-Bank 058

Verify that the points $(1\ ,\ 1)$ and $(-2 ,\ 7)$ lie on the line $y=-2x+3$? (3 marks)

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$\text{See worked solution}$

Show Worked Solution

$\text{Check points by substituting into }y=-2x+3$

$(1\ ,\ 1) \longrightarrow$	$RHS$	$=-2\times 1+3$
		$=1$
		$=LHS$

$(-2 ,\ 7) \longrightarrow$	$RHS$	$=-2\times (-2)+3$
		$=7$
		$=LHS$

$\therefore\ (1\ ,\ 1)\ \text{and }(-2 ,\ 7) \text{ both lie on the line}\ \ y=-2x+3$

Linear Relationships, SM-Bank 057

Verify that the points $(2\ ,\ 5)$ and $(-1 ,-1)$ lie on the line $y=2x+1$? (3 marks)

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$\text{See worked solution}$

Show Worked Solution

$\text{Check points by substituting into }y=2x+1$

$(2\ ,\ 5) \longrightarrow$	$RHS$	$=2\times 2+1$
		$=5$
		$=LHS$

$(-1 ,-1) \longrightarrow$	$RHS$	$=2\times (-1)+1$
		$=-1$
		$=LHS$

$\therefore\ (2\ ,\ 5)\ \text{and }(-1 ,-1) \text{ both lie on the line}\ \ y=2x+1$

Linear Relationships, SM-Bank 056 MC

Which of the following points lies on the line $y=2x-4$?

$(0 , 2)$
$(-2 , 8)$
$(2 , -1)$
$(-1, -6)$

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$D$

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$\text{Check each option by substituting into }y=2x-4$

$\text{Option A:}\ \ \ $	$2$	$\ne 2\times 0-4=-4$
$\text{Option B:}$	$8$	$\ne 2\times (-2)-4=-8$
$\text{Option C:}$	$-1$	$\ne 2\times 2-4=0$
$\text{Option D:}$	$-6$	$=2\times (-1)-4=-6\ \ \ \checkmark$

$\therefore\ (-1, -6) \text{ lies on the line}\ \ y=2x-4$

$\Rightarrow D$

Linear Relationships, SM-Bank 055 MC

Which of the following points lies on the line $y=10+x$?

$(3 , 7)$
$(-2 , 8)$
$(2 , -8)$
$(-6, -4)$

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$B$

Show Worked Solution

$\text{Check each option by substituting into }y=10+x$

$\text{Option A:}\ \ \ $	$7$	$\ne 10+3$
$\text{Option B:}$	$8$	$=10+-2\ \ \ \checkmark$
$\text{Option C:}$	$-8$	$\ne 10+2$
$\text{Option D:}$	$-4$	$\ne 10+-6$

$\therefore\ (-2, 8) \text{ lies on the line}\ \ y=10+x$

$\Rightarrow B$

Linear Relationships, SM-Bank 054

Renee and Leisa are saving money so they can visit their grandmother on a holiday.

Renee has $100 and plans to save $30 each week.

Leisa has $200 and plans to save $10 each week.

Write an equation to represent
(i) Renee's savings (1 mark)

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(ii) Leisa's savings (1 mark)

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Complete the following tables of values for the equations above. (2 marks)

Renee's savings
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Weeks }(w) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Savings }(s) \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}

$\ \ \ \ \ \ \ $

Leisa's savings
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Weeks }(w) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Savings }(s) \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}

Using the tables of values, graph both equations on the number plane below. Be sure to extend your lines to the end of the grid. (2 marks)
After how many weeks will Renee and Leisa have saved the same amount? (1 mark)

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a. (i) $s=100+30w$

(ii) $s=200+10w$

$\text{Renee’s savings: }s=100+30w$
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Weeks }(w) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Savings }(s) \ \rule[-1ex]{0pt}{0pt} & 100 & 130 & 160 & 190\\
\hline
\end{array}

$\ \ \ \ \ \ \ $

$\text{Leisa’s savings: }s=200+10w$
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Weeks }(w) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Savings }(s) \ \rule[-1ex]{0pt}{0pt} & 200 & 210 & 220 & 230\\
\hline
\end{array}

d. $5\ \text{weeks}$

Show Worked Solution

a. (i) $s=100+30w$

(ii) $s=200+10w$

$\ \ \ \ \ \ \ $

d. $\text{Method 1 – Graphically by inspection}$

$\text{Lines intersect when }w=5\ \text{and }s=$250$

$\text{Method 2 – Algebraically}$

$\text{Solve }s=100+30w\ \text{ and }s=200+10w\ \text{simultaneously}$

$100+30w$	$=200+10w$
$30w-10w$	$=200-100$
$20w$	$=100$
$w$	$=\dfrac{100}{20}=5$

$\therefore\ \text{Amounts are equal after }5 \text{ weeks}.$

Linear Relationships, SM-Bank 053 MC

Paddy and Miffy each bought potatoes from the local farmers' market.

Paddy bought 2 kilograms for $2.50.

Miffy bought 6 kilograms for $7.50.

Which graph best represents the cost of potatoes at the farmers' market?

A.	B.

C.	D.

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$D$

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$\Rightarrow D$

Linear Relationships, SM-Bank 052

Jeremy owns a paddle board hire company. He charges a $20 insurance fee with every hire and $35 for every hour of hire.

Complete the table of values below, where $\large h$ represents the number of hours of hire and $\large w$ represents his total earnings for each hour of hire. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Hours }(h) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 2\ \ &\ \ 4\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Wage }(w) \ \rule[-1ex]{0pt}{0pt} & & & \\
\hline
\end{array}
Write an equation to represent Jeremy's wages, using the variables $h$ and $w$. (1 mark)

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Graph the equation of the line representing Jeremy's wages on the number plane below. Be sure to extend your line to the edge of the grid. (2 marks)
For how many hours would Jeremy have to hire a paddle board to earn $230? (1 mark)

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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Hours }(h) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 2\ \ &\ \ 4\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Wage }(w) \ \rule[-1ex]{0pt}{0pt} & 20 & 90 & 160 \\
\hline
\end{array}

b. $w=20+35h$

d. $6\ \text{hours}$

Show Worked Solution

a. $\text{Table of values}$

b. $w=20+35h$

d. $\text{Method 1 – Graphically by inspection}$

$\text{When }w=230 , h=6\ \text{hours}$

$\text{Method 2 – Algebraically}$

	$w$	$=20+35h$
	$230$	$=20+35h$
	$35h$	$=230-20$
	$35h$	$=210$
	$h$	$=\dfrac{210}{35}=6$

$\therefore\ \text{Jeremy would have to hire a board for }6\text{ hours to earn } $230.$

Linear Relationships, SM-Bank 051

Julie cleans carpets and upholstery. She charges a $40 call-out fee and $20 for every hour it takes to complete a job.

Complete the table of values below, where $\large h$ represents the number of hours Julie works and $\large w$ represents her total wage. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Hours }(h) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Wage }(w) \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}
Write an equation to represent Julie's wages, using the variables $h$ and $w$. (1 mark)

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Graph the equation of the line representing Julie's wages on the number plane below. Be sure to extend your line to the edge of the grid. (2 marks)
For how many hours would Julie have to clean to earn $150? (1 mark)

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\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\text{Hours }(h) \ \rule[-1ex]{0pt}{0pt} &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ &\ \ 3\ \ \\
\hline
\rule{0pt}{2.5ex} \text{Wage }(w) \ \rule[-1ex]{0pt}{0pt} & 40 & 60 & 80 & 100 \\
\hline
\end{array}

b. $w=40+20h$

d. $5.5\ \text{hours}$

Show Worked Solution

a. $\text{Table of values}$

b. $w=40+20h$

d. $\text{Method 1 – Graphically by inspection}$

$\text{When }w=150 , h=5.5\ \text{hours}$

$\text{Method 2 – Algebraically}$

	$w$	$=40+20h$
	$150$	$=40+20h$
	$20h$	$=150-40$
	$20h$	$=110$
	$h$	$=\dfrac{110}{20}=5.5$

$\therefore\ \text{Julie would have to work for }5.5\text{ hours to earn } $150.$

Linear Relationships, SM-Bank 050 MC

Which of the following is not true of the lines on the number plane below?

$\text{They have the same }y\text{-intercept}$.
$\text{They both pass through the point}\ (0,-1)$.
$\text{They constant value in the equations of both lines is }-1$.
$\text{They both pass through the point}\ (-1,0)$.

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$D$

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$\text{Neither of the lines pass through the point }(-1,0).$

$\Rightarrow D$

Linear Relationships, SM-Bank 049 MC

What do the lines on the following number plane have in common?

$\text{They have the same }y\text{-intercept}$.
$\text{They are parallel to each other}$.
$\text{They both pass through the origin}$.
$\text{They are perpendicular to each other}$.

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$B$

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$\text{The lines are parallel}.$

$\Rightarrow B$

Linear Relationships, SM-Bank 048 MC

What do all the lines on the following number plane have in common?

$\text{They all intersect at the point }(2,1)$.
$\text{They are all parallel to each other}$.
$\text{They all intersect at the point }(1,2)$.
$\text{They are all perpendicular to each other}$.

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$C$

Show Worked Solution

$\text{The lines all pass through the point }(1,2).$

$\therefore\ \text{They all intersect at the point }(1,2)$.

$\Rightarrow C$

Linear Relationships, SM-Bank 047

Complete the tables of values below for each given rule. (3 marks)

$y=3-x$
\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}

$y=3x-1$
\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}

On the number plane below, graph the equations from part (a). (2 marks)
Using the graph, find the point of intersection of the two lines. (1 mark)

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c. $(1 , 2)$

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c. $\text{Point of intersection: }(1 , 2)$

Linear Relationships, SM-Bank 046 MC

Monique has correctly drawn the graph of $y=-2x+5$ on the number plane below.

She used the points $(-1,7),\ (0,5)$ and $(1,3)$ to draw the line.

How many more different points could she have used to plot the line $y=-2x+5$?

$0$
$2$
$8$
$\text{An infinite number}$

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$D$

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$\text{Straight lines are made up of an infinite number of points.}$

$\Rightarrow D$

Linear Relationships, SM Bank 045

Complete the tables of values below for each given rule. (3 marks)

$y=2x+1$
\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}

$y=x-2$
\begin{array} {|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & & & & \\
\hline
\end{array}

On the number plane below, graph the equations from part (a). (2 marks)
Using the graph, find the point of intersection of the two lines. (1 mark)

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c. $(-3 , -5)$

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c. $\text{Point of intersection: }(-3 , -5)$

Linear Relationships, SM-Bank 044

Complete the table of values using the graph of the straight line below. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & & & & & \\
\hline
\end{array}
Using either the table or the graph, state the rule connecting $\large x$ and $\large y$. (1 mark)

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\begin{array} {|l|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 3 & 2 & 1 & 0 & -1 \\
\hline
\end{array}

b. $y=-x+1$

Show Worked Solution

a. $\text{Table of values}$

b. $\text{The }y\ \text{values are decreasing by } 1\ \text{and when }x=0,\ \ y=1$

$\therefore\ \text{Rule: }y=-x+1$

Linear Relationships, SM-Bank 043

Complete the table of values using the graph of the straight line below. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & & & & & \\
\hline
\end{array}
Using either the table or the graph, state the rule connecting $\large x$ and $\large y$. (1 mark)

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b. $y=2x+1$

Show Worked Solution

a. $\text{Table of values}$

b. $\text{The }y\ \text{values are increasing by } 2\ \text{and when }x=0,\ \ y=1$

$\therefore\ \text{Rule: }y=2x+1$

Linear Relationships, SM-Bank 042

Plot the points from the table on the number plane below and join the points using a ruler. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} & -2 & -1 &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 2 & 2 & 2 & 2 & 2\\
\hline
\end{array}
What do you notice about the points? (1 mark)

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Using either the table or the graph, state the rule connecting $\large x$ and $\large y$. (1 mark)

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b. $\text{They form a horizontal straight line.}$

c. $y=2$

Show Worked Solution

b. $\text{They form a horizontal straight line.}$

\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} &\ \ -2\ \ &\ \ -1\ \ &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 2 & 2 & 2 & 2 & 2\\
\hline
\end{array}

$y=2\text{ regardless of the value of }x$

$\therefore\ \text{Rule: }y=2$

Linear Relationships, SM-Bank 041

Plot the points from the table on the number plane below and join the points using a ruler. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} &\ \ -2\ \ &\ \ -1\ \ &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & 5 & 4 & 3 & 2 & 1\\
\hline
\end{array}
What do you notice about the points? (1 mark)

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Using either the table or the graph, state the rule connecting $\large x$ and $\large y$. (1 mark)

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b. $\text{They form a straight line.}$

c. $y=3-x$

Show Worked Solution

b. $\text{They form a straight line.}$

$\therefore\ \text{Rule: }y=3-x$

Linear Relationships, SM-Bank 040

Plot the points from the table on the number plane below and join the points using a ruler. (2 marks)
\begin{array} {|l|c|c|c|}
\hline
\rule{0pt}{2.5ex}\ \ x\ \ \rule[-1ex]{0pt}{0pt} &\ \ -2\ \ &\ \ -1\ \ &\ \ 0\ \ &\ \ 1\ \ &\ \ 2\ \ \\
\hline
\rule{0pt}{2.5ex} \ \ y\ \ \rule[-1ex]{0pt}{0pt} & -4 & -2 & 0 & 2 & 4\\
\hline
\end{array}
What do you notice about the points? (1 mark)

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Using either the table or the graph, state the rule connecting $\large x$ and $\large y$. (1 mark)

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b. $\text{They form a straight line.}$

c. $y=2x$

Show Worked Solution

b. $\text{They form a straight line.}$

$\therefore\ \text{Rule: }y=2x$

\((1\ ,-1) \longrightarrow\)	\(LHS\)	\(=1+2\times (-1)\)
		\(=1-2=-1\)
		\(=RHS\)

\((-7 ,\ 3) \longrightarrow\)	\(LHS\)	\(=-7+2\times 3\)
		\(=-7+6=-1\)
		\(=RHS\)

\((1\ ,\ 1) \longrightarrow\)	\(RHS\)	\(=-2\times 1+3\)
		\(=1\)
		\(=LHS\)

\((-2 ,\ 7) \longrightarrow\)	\(RHS\)	\(=-2\times (-2)+3\)
		\(=7\)
		\(=LHS\)

\((2\ ,\ 5) \longrightarrow\)	\(RHS\)	\(=2\times 2+1\)
		\(=5\)
		\(=LHS\)

\(\text{Option A:}\ \ \ \)	\(2\)	\(\ne 2\times 0-4=-4\)
\(\text{Option B:}\)	\(8\)	\(\ne 2\times (-2)-4=-8\)
\(\text{Option C:}\)	\(-1\)	\(\ne 2\times 2-4=0\)
\(\text{Option D:}\)	\(-6\)	\(=2\times (-1)-4=-6\ \ \ \checkmark\)

\(\text{Option A:}\ \ \ \)	\(7\)	\(\ne 10+3\)
\(\text{Option B:}\)	\(8\)	\(=10+-2\ \ \ \checkmark\)
\(\text{Option C:}\)	\(-8\)	\(\ne 10+2\)
\(\text{Option D:}\)	\(-4\)	\(\ne 10+-6\)

\(100+30w\)	\(=200+10w\)
\(30w-10w\)	\(=200-100\)
\(20w\)	\(=100\)
\(w\)	\(=\dfrac{100}{20}=5\)

	\(w\)	\(=20+35h\)
	\(230\)	\(=20+35h\)
	\(35h\)	\(=230-20\)
	\(35h\)	\(=210\)
	\(h\)	\(=\dfrac{210}{35}=6\)