Linear Graphs (OCR GCSE Maths: Higher): Exam Questions

On the grid, draw the graph of $y=2x-3$ for values of $x$ from $-2\mathrm{to}3$

On the grid, draw the graph of $y=3x+2$ for values of $x$ from $-2\mathrm{to}2$

The equation of the line ${\mathrm{L}}_1$ is $y=3x-2$
The equation of the line ${\mathrm{L}}_2$ is $3y-9x+5=0$

Show that these two lines are parallel.

${\mathbf{L}}_1$ and ${\mathbf{L}}_2$ are parallel lines.

The equation of ${\mathbf{L}}_1$ is $y=3x+2$
${\mathbf{L}}_2$ passes through the point  $(3,4)$.

Find an equation for ${\mathbf{L}}_2$.

The diagram shows a straight line, ${\mathrm{L}}_1$, drawn on a grid.

A straight line, ${\mathrm{L}}_2$, is parallel to the straight line ${\mathrm{L}}_1$ and passes through the point $(0,-5)$.
Find an equation of the straight line ${\mathrm{L}}_2$.

The line $l_1$ has equation $3x+5y-2=0$

Find the gradient of $l_1$.

The line $l_2$ is perpendicular to $l_1$ and passes through the point (3, 1).

Find the equation of $l_2$ in the form $y=mx+c$, where $m$ and $c$ are constants.

The graph shows two parallel lines, Line A and Line B.

Line A has equation $y=6x+7$.
Line B passes through the point (4, 26).

Find the equation of Line B.

Point A has coordinates (-4, 6) and point B has coordinates (8, 3).

i) Find the gradient of line AB.

[2]

ii) Find the equation of line AB.

[2]

Point P has coordinates (0, -2).

Write down the equation of the line parallel to line AB that passes through P.

Show that line 3y = 4x − 14 is perpendicular to line 4y = −3x + 48.

A straight line passes through the point (0, 6) and is perpendicular to  y = 4x − 5.

Find the equation of this line, giving your answer in the form y = mx + c.

The straight line ${\mathbf{L}}_{\mathbf{1}}$ has equation $x+2y=4$

The straight line ${\mathbf{L}}_{\mathbf{2}}$ passes through the points $(-1,-7)$ and $(7,9)$
Michael says that the lines ${\mathbf{L}}_{\mathbf{1}}$ and ${\mathbf{L}}_{\mathbf{2}}$ are perpendicular.

Is Michael correct?
You must show clearly how you get your answer.

The straight line ${\mathbf{L}}_{\mathbf{1}}$has equation $2y=6x-5$

The straight line ${\mathbf{L}}_{\mathbf{2}}$ is perpendicular to ${\mathbf{L}}_{\mathbf{1}}$ and passes through the point (9 , -1)

Find an equation for ${\mathbf{L}}_{\mathbf{2}}$
Give your answer in the form $ay+bx=c$

The equation of a straight line is    $3x+2y=24$

Circle the point where the line crosses the $x$-axis.

A straight line

has gradient 6
and
passes through the point (3, 19)

Work out the equation of the line. Give your answer in the form $y=mx+c$

Jim buys a plant of height 20 cm.

The graph shows how the height of the plant changes during the next 4 days.

Work out a formula for$h$ in terms of $n$.

$AB$is a line segment.

The midpoint of the line segment $AB$has coordinates (3, 5) Point $A$has coordinates (9, 2)

Work out the coordinates of point $B$.

Work out an equation of the straight line that passes through (9, 2) and (3, 5)

The line $\mathbf{N}$ is drawn below.

Find an equation of the line perpendicular to line $\mathbf{N}$ that passes through the point (0, 1).

The straight line $\mathbf{L}$ has equation $4x+y=7$

Find an equation of the straight line perpendicular to $\mathbf{L}$ that passes through (-8, 3).

The straight line $\mathbf{L}$ has the equation $3y=4x+7$ The point $A$ has coordinates (3, —5)

Find an equation of the straight line that is perpendicular to $\mathbf{L}$ and passes through $A$.

Find an equation of the line that passes through $\mathrm{C}$ and is perpendicular to $AB$.

The line $L$ has equation $y=5-2x.$

Show that the point $P(3,—1)$ lies on $L$.

Find an equation of the line perpendicular to $L$, which passes through $P$. Give your answer in the form $ax+by+c=0$, where $a,b$ and $c$ are integers.

The line $l_1$ has equation $y=-2x+3$
The line $l_2$ is perpendicular to $l_1$ and passes through the point $(5,6)$.

Find an equation for $l_2$ in the form $ax+by+c=0$, where $a,b$ and $c$ are integers.

The line $l_2$ crosses the $x$-axis at the point $A$ and the $y$-axis at the point $B$.

Find the $x$-coordinate of $A$ and the $y$-coordinate of $B$.

Given that $O$ is the origin,

find the area of the triangle $OAB$.

$A$ is the point with coordinates $(1,3)$

$B$ is the point with coordinates $(4,-1)$
The straight line $L$ goes through both $A$ and $B$.

Is the line with equation $2y=3x—4$ perpendicular to line $L$? You must show how you got your answer.

P is the point (0, -1) and Q is the point (5, 9).

Find the equation of the line through P that is perpendicular to the line PQ.

The straight line $\mathbf{L}$ passes through the points (4, −1) and (6, 4)

The straight line $\mathbf{M}$ is perpendicular to $\mathbf{L}$ and intersects the $y$-axis at the point (0, 8)

Find the coordinates of the point where $\mathbf{M}$ intersects the $x$-axis.

$ABCD$ is a rhombus.

The diagonals, $AC$ and $BD$, intersect at the point $M$. The coordinates of $M$ are $(6,-11)$

The points $A$ and $C$ both lie on the line with equation $2y+7x=20$

Find the exact coordinates of the point where the line through $B$ and $D$ intersects the $y-\mathrm{axis}$.

Here is line L and the graph of $y=x–1$

The scales of the axes are not shown.

Work out the equation of line L.

Show that the lines $y=3x+7$ and $2y–6x=8$ are parallel.

Do not use a graphical method.

Is the point (–5, –6) above, below or on the line $y=3x+7$ ?

Tick one box.

$\square$ Above   $\square$ Below   $\square$On the line

You must show your working.
Do not use a graphical method.

A straight line passes through the points $(0,4)$ and $(2,0)$.

A second straight line is parallel to the first line and passes through the point $(5,1)$.

Find the equation of the second straight line.

Give your answer in the form $y=mx+c$.

You must show your working.

The points $P$ and $Q$ have coordinates $\left(-1,6\right)$ and $\left(9,0\right)$ respectively. 

The line $l$ is perpendicular to $PQ$ and passes through the midpoint of $PQ.$

Find an equation for $l.$

Figure 1

The line $l_1$ has equation $2x-3y+12=0$

Find the gradient of $l_1$.

The line $l_1$ crosses the $x$-axis at the point $A$and the $y$-axis at the point $B$, as shown in Figure 1.

The line $l_2$ is perpendicular to $l_1$ and passes through $B$.

Find an equation of $l_2$.

The line $l_2$ crosses the $x$-axis at the point $C$.

Find the area of triangle $ABC$.

$ABCD$ is a rhombus.
The coordinates of $A$ are $\left(5,11\right)$
The equation of the diagonal $DB$ is $y=\frac{1}{2}x+6$
Find an equation of the diagonal $AC$.

Figure 2

The line $l_1$, shown in Figure 2 has equation $2x+3y=26$ The line $l_2$ passes through the origin $O$ and is perpendicular to $l_1$

Find an equation for the line $l_2$

The line $l_2$ intersects the line $l_1$ at the point $C$.
Line $l_1$ crosses the $y$-axis at the point $B$ as shown in Figure 2.

Find the area of triangle $OBC$.

Give your answer in the form $\frac{a}{b}$, where $a$ and $b$ are integers to be determined.

The point $P$ has coordinates $(3,4)$
The point $Q$ has coordinates $(a,b)$

A line perpendicular to $PQ$ is given by the equation $3x+2y=7$

Find an expression for $b$ in terms of $a$.

$ABCD$ is a square.
$P$ and $D$ are points on the $y$-axis.
$A$ is a point on the $x$-axis.
$PAB$ is a straight line.

The equation of the line that passes through the points $A$ and $D$ is $y=—2x+6$
Find the length of $PD$.

$A$(—2, 1), $B$(6, 5) and $C$(4, $k$) are the vertices of a right-angled triangle $ABC$. Angle $ABC$ is the right angle.

Find an equation of the line that passes through $A$ and $C$. Give your answer in the form $ay+bx=c$ where $a$, $b$ and $c$are integers.

The points $A$ and $B$ have coordinates (3, 4) and (7, —6) respectively. The straight line $l$ passes through $A$ and is perpendicular to $AB$. Find an equation for $l$, giving your answer in the form $ax+by+c=0$, where $a,b$ and $c$ are integers.

$ABCD$is a rectangle.

$A,E$ and $B$ are points on the straight line $\mathbf{L}$with equation $x+2y=12$ $A$ and $D$ are points on the straight line $\mathbf{M}$.

$AE=EB$

Find an equation for $\mathbf{M}$.

${\mathbf{L}}_{\mathbf{1}}$and ${\mathbf{L}}_{\mathbf{2}}$ are two straight lines.
The origin of the coordinate axes is $O$.

${\mathbf{L}}_{\mathbf{1}}$ has equation $5x+10y=8$
${\mathbf{L}}_{\mathbf{2}}$ is perpendicular to ${\mathbf{L}}_{\mathbf{1}}$ and passes through the point with coordinates (8, 6)

${\mathbf{L}}_{\mathbf{2}}$ crosses the $x$-axis at the point $A$.
${\mathbf{L}}_{\mathbf{2}}$ intersects the straight line with equation $x=–3$ at the point $B$.

Find the area of triangle $AOB$.
Show your working clearly.

Line A has equation $y=4x–1$

Line B is

perpendicular to line A
and
passes through the point (8, 5)

Work out the coordinates of the point where line B intersects the $x$-axis.