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#### a)

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Parallel lines are defined as lines that remain the same distance apart and so they never meet. This means that their gradient is the same.

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If we write our line equations in the form $y_i=m_i\\times x_i+c_i$, then for parallel lines, $m_1=m_2$.

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Vertical lines are always parallel to the $y$-axis and horizontal lines are always parallel to the $x$-axis.

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#### b)

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All but one of the straight lines above follow the general formula $y=mx+c$ where $m$ is a constant denoting the gradient of the line and $c$ is a constant denoting the $y$-intercept of the line.

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The gradient (given by $m$), indicates the slope of a straight line. Where this is positive, the $y$-coordinate along the line increases by $m$ for every increase in the value of $x$ by $1$. For example, $y=3x$ indicates that for every increase in the value of $x$ by $1$, the value of $y$ increases by $3$. Where there is no $m$ (i.e. $y=c$), the line is horizontal as the value of $y$ has no dependence on the value of $x$.

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The $y$-intercept (given by $c$), indicates the point where the line passes through the $y$-axis. If the line equation has no $c$ (i.e. $y=mx$), the line has a $y$ intercept of $0$ (passing through the origin).

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We can use our knowledge of the slope of the line and the $y$-intercept to then judge which points on the graph correspond to which equations.

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A:

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$y=\\simplify{{m1}x+{c1}}$  gives us $m=\\var{m1}$ and $c=\\var{c1}$.

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B:

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$y=\\simplify{{m2}x+{c2}}$  gives us $m=\\var{m2}$ and $c=\\var{c2}$.

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C:

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$y=\\simplify{{m3}x+{c3}}$  gives us $m=\\var{m3}$ and $c=\\var{c3}$.

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D:

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$y=\\simplify{{c4}}$  gives us $m=0$ and $c=\\var{c1}$.

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E:

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$x=\\var{c5}$  is the exception to the $y=mx+c$ form of writing straight line equations. This equation gives a value for $x$ independent of the value of $y$, so a vertical line is formed at the $x$-coordinate $\\var{c5}$.

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#### c)

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For the equation $y=\\var{cfake}$, the gradient is $0$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find $D_0$ has the same gradient. Therefore $D_0$ is parallel to $y=\\var{cfake}$.

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#### d)

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For the equation $y=\\var{m1}x+\\var{cpar}$, the gradient is $\\var{m1}$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find both $A_0$ and $B_0$ have the same gradient. Therefore $A_0$ and $B_0$ are parallel to $y=\\var{m1}x+\\var{cpar}$.

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#### e)

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This time, we have to find the gradient of the $x$-axis. Note that the $x$-axis is equivalent to the line with equation $y=0$. The gradient of this line is $0$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find $D_0$ has the same gradient. Therefore $D_0$ is parallel to $y=0$ and in other words, $D_0$ is parallel to the $x$-axis.

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#### f)

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For the equation $y=\\simplify{{muniq}x+{c1}}$, the gradient is $\\var{muniq}$, so we check our lines $A_0, B_0, C_0, D_0 \\text{ and }E_0$ to find none have the same gradient. Therefore none of the labelled lines are parallel to $y=\\simplify{{muniq}x+{c1}}$.

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#### g)

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Lines that are parallel have the same gradient. This is represented by $m$ in the general form of a straight line ($y=mx+c$). The gradient of $C_0$ was $\\var{m3}$, so any straight line parallel to $C_0$ will share this value for $m$.

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Therefore, we input $y=\\var{m3}x+c$.

", "statement": "

Use the graph bellow to wok out the line equations of lines $A_0 \\text{ to } E_0$ and use these equations to judge which lines they are parallel to.

\n

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Some conceptual questions about parallel lines - fill in the gaps in some statements.

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Identify the lines corresponding to given equations, and the lines parallel to lines with given equations.

\n

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the same

", "

a variable

", "

a large

", "

a small

", "

a negligible

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", "

$y$-intercept

", "

length

", "

color

", "

$y$-coordinate

", "

$x$-coordinate

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m\u2081

", "

c\u2081

", "

x\u2081

", "

y\u2081

"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["1", 0, 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "type": "1_n_2", "choices": ["

m\u2082

", "

c\u2082

", "

x\u2082

", "

y\u2082

"], "showFeedbackIcon": true, "minMarks": 0}, {"variableReplacements": [], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "matrix": ["0", "1", 0, 0], "scripts": {}, "displayColumns": 0, "showCorrectAnswer": true, "distractors": ["", "", "", ""], "shuffleChoices": false, "marks": 0, "type": "1_n_2", "choices": ["

the x-axis

", "

the y-axis

", "

y=x

", "

nothing

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the x-axis

", "

the y-axis

", "

y=x

", "

nothing

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Fill in the gaps below to complete the description of parallel lines.

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Parallel lines are defined as a lines that remain [[0]] distance apart and so they never meet. This means that their [[1]] is the same.

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If we write our line equations in the form $y=m_i\\times x_i+c_i$, then for parallel lines, [[2]] $=$ [[3]].

\n

Vertical lines are always parallel to [[4]] and horizontal lines are always parallel to [[5]].

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$A_0$

", "

$B_0$

", "

$C_0$

", "

$D_0$

", "

$E_0$

", "

None of these

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$y=\\simplify{{m1}x+{c1}}$

", "

$y=\\simplify{{m2}x+{c2}}$

", "

$y=\\simplify{{m3}x+{c3}}$

", "

$y=\\var{c4}$

", "

$x=\\var{c5}$

", "

$y=\\simplify{{mfake}x+{cfake}}$

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Use the graph above to match the correct line to each equation.

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Which lines are parallel to the line with equation $y=\\var{cfake}$?

", "type": "m_n_2", "choices": ["

$A_0$

", "

$B_0$

", "

$C_0$

", "

$D_0$

", "

$E_0$

", "

None of these

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Which lines are parallel to the line with equation $y=\\var{m1}x+\\var{cpar}$?

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$A_0$

", "

$B_0$

", "

$C_0$

", "

$D_0$

", "

$E_0$

", "

None of these

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Which lines are parallel to the $x$ axis?

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$A_0$

", "

$B_0$

", "

$C_0$

", "

$D_0$

", "

$E_0$

", "

None of these

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Which lines are parallel to the line with the equation $y=\\simplify{{muniq}x+{c1}}$?

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$A_0$

", "

$B_0$

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$C_0$

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$D_0$

", "

$E_0$

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None of these

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Complete the general equation in the form $y=mx+c$ for any line parallel to line $C_0$.

\n

$y=$[[0]]$x+c$

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