Topics
4. Systems of Equations
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Standard Level

4.02 Graphical method

Lesson
Worksheet
Practice
Worksheet
Graphical solutions
1

Each graph below shows a system of two equations. Write down the solution to each system, as a point in the form \left(x, y\right).

a
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
b
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
c
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
d
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
e
-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
2

Consider the graph showing a system of two equations:

a

How many solutions does this system of equations have?

b

Write down the solution to the system of equations as an ordered pair \left(x, y\right).

-6
-4
-2
2
4
6
x
-6
-4
-2
2
4
6
y
3

Consider the graph showing a system of two equations:

How many solutions does this system of equations have?

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
4

Consider the graph of the equation \\y = 5 x + 3:

If a second line y = mx + b intersects this line at the point \left(0, 3\right), state the value of b.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
5

Consider the following systems of linear equations:

i

Find the x and y-intercepts of the first equation.

ii

Find the x and y-intercepts of the second equation.

iii

Graph the two equations on the same set of axes.

iv

State the coordinate pair \left(x, y\right) that satisfies both equations.

a
\begin{aligned} y &= x - 1 \\ y &= - 2 x + 8 \end{aligned}
b
\begin{aligned} y &= x + 1 \\ y &= 4 x + 4 \end{aligned}
c
\begin{aligned} y &= x + 3 \\ y &= 3 x + 3 \end{aligned}
d
\begin{aligned} y &= -2x + 2 \\ y &= x − 7 \end{aligned}
e
\begin{aligned} y &= -x - 1 \\ y &= -3x - 3 \end{aligned}
f
\begin{aligned} y &= 2x + 2 \\ y &= -x + 2 \end{aligned}
6

Consider the following linear equations:

\begin{aligned} y &= 2 x + 2 \\ y &= - 2 x + 2 \end{aligned}

a

Determine the gradient and y-intercept of the line y = 2 x + 2.

b

Determine the x\text{ and }y-intercepts of the line y = - 2 x + 2.

c

Hence sketch the graphs on the same coordinate plane.

d

State the values of x and y which satisfy both equations.

7

Consider the following system of equations:

\begin{aligned} y &= 4 x - 3 \\ y &= 4 - 3 x \end{aligned}

a

Complete the table of values for y = 4 x - 3:

x-3-2-10123
y
b

Complete the table of values for y = 4 - 3 x:

x-3-2-10123
y
c

Find the gradient of y = 4 x - 3.

d

Find the gradient of y = 4 - 3 x.

e

Graph the two equations on the same set of axes.

f

Using your answer from part (e), find the point of intersection of the lines.

8

Consider the following system of equations:

\begin{aligned} y &= x - 2 \\ y &= - 3 x - 6 \end{aligned}
a

Find the gradient, and the y-value of the y-intercept, of the line y = x - 2.

i

gradient

ii

y-value of the y-intercept

b

Find the gradient, and the y-value of the y-intercept, of the line y = - 3 x - 6.

i

gradient

ii

y-value of the y-intercept

c

Graph the two equations on the same set of axes.

d

State the coordinate pair \left(x, y\right) that satisfies both equations.

9

Consider the following system of equations:

\begin{aligned} y &= - 4 x - 1 \\ y &= - 4 x + 2 \end{aligned}

a

Graph the two equations on the same set of axes.

b

Does there exist a value for x and y that satisfy the following two equations simultaneously?

10

For each of the following systems of equations:

i

Graph the two equations on the same set of axes.

ii

State the values of x and y that satisfy both equations.

a
\begin{aligned} y &= x \\ y &= - 1 \end{aligned}
b
\begin{aligned} 4 x - 2 y &= 2 \\ - 2 x + 4 y &= 2 \end{aligned}
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