Exercises: graphically solving systems of equations

Solving systems of equations graphically means reading off intersection points from a diagram.

1
Determining intersection points
Consider two straight lines g and h.
Determine the linear equations of g and h.

$y = m x + t$
The parameter $m$ is the slope of the line. The parameter $t$ is the y-intercept.
Line g:
$y = m_{g} x + t_{g}$
The straight line g has slope $- \frac{3}{2}$ , that is $m_{g} = - \frac{3}{2}$ .
$y = - \frac{3}{2} x + t_{g}$
It intersects the y-axis at $\frac{9}{2}$ , that is $t_{g} = \frac{9}{2}$ .
$y = - \frac{3}{2} x + \frac{9}{2}$
Line h:
$y = m_{h} x + t_{h}$
The straight line h has slope 2, that is $m_{h} = 2$
$y = 2 x + t_{h}$
The straight line intersects the y-axis at 1, that is $t_{h} = 1$ .
$y = 2 x + 1$

Read off the intersection point.

Reading off intersection points
We read off the intersection point $IP (1 | 3)$ from the drawing.
2
Determining intersection points
Three straight lines are drawn in the diagram below. Read off the intersection points from the figure. .

For this task you need the following basic knowledge: System of linear equations
Reading off intersectio points
In the diagram on the right you can determine the approximate coordinates of the three points.
3
Decide whether the following systems of linear equations are solvable or not. Make a sketch of the corresponding linear functions.
$\begin{array}{rcl} (I) & x & + & y & = & 5 \\ (II) & - 2 x & + & 3 y & = & 6 \end{array}$
has exactly one solution
has no solution
has infinitely many solutions

For this task you need the following basic knowledge: System of linear equations
$(I)$ solve for $y$
$\begin{array}{rcll} x + y & = & 5 & | - x \\ y & = & - x + 5 \end{array}$
$(II)$ solve for $y$
$\begin{array}{rcll} - 2 x + 3 y & = & 6 & | + 2 x \\ 3 y & = & 2 x + 6 & | : 3 \\ y & = & \frac{2}{3} x + 2 \end{array}$
Draw both straight lines in a coordinate system
Answer
The linear system of equations has exactly one solution, since the two straight lines intersect at the point $S$ .
First, solve the equations $(I)$ and $(II)$ for $y$ and draw the corresponding straight lines into a coordinate system.

$\begin{array}{rcl} (I) & 8 x & + & 2 y & = & 16 \\ (II) & 16 x & + & 4 y & = & 32 \end{array}$
has exactly one solution
has no solution
has infinitely many solutions

For this task you need the following basic knowledge: System of linear equations
$(I)$ solve for $y$
$\begin{array}{rcll} 8 x + 2 y & = & 16 & | - 8 x \\ 2 y & = & - 8 x + 16 & | : 2 \\ y & = & - 4 x + 8 \end{array}$
$(II)$ solve for $y$
$\begin{array}{rcll} 16 x + 4 y & = & 32 & | - 16 x \\ 4 y & = & - 16 x + 32 & | : 4 \\ y & = & - 4 x + 8 \end{array}$
Draw both straight lines in a coordinate system (both lines are the same)
Answer
The linear system of equations has infinitely many solutions.
The equations (\text{I}) and (\text{II}) describe the same linear function. Therefore, for each x\text{I} (x|y)=(x|-4x+8) is a solution of the linear system of equations.
First solve the equations $(I)$ and $(II)$ for $y$ and draw the corresponding straight lines into a coordinate system.

$\begin{array}{rcl} (I) & 5 x & - & 2 y & = & 2 \\ (II) & 15 x & - & 6 y & = & - 24 \end{array}$
has exactly one solution
has no solution
has infinitely many solutions

For this task you need the following basic knowledge: System of linear equations
$(I)$ solve for $y$
$\begin{array}{rcll} 5 x - 2 y & = & 2 & | + 2 y - 2 \\ 5 x - 2 & = & 2 y \\ 2 y & = & 5 x - 2 & | : 2 \\ y & = & \frac{5}{2} x - 1 \end{array}$
$(II)$ solve for $y$
$\begin{array}{rcll} 15 x - 6 y & = & - 24 & | + 6 y + 24 \\ 15 x + 24 & = & 6 y \\ 6 y & = & 15 x + 24 & | : 6 \\ y & = & \frac{15}{6} x + 4 \\ y & = & \frac{5}{2} x + 4 \end{array}$
Draw both straight lines in a coordinate system (both lines are parallel, as they have the same slope)
Answer
The linear system of equations has no solution
The corresponding straight lines resulting from the linear system of equations are parallel to each other and do not intersect.
First solve the equations $(I)$ and $(II)$ for $y$ and draw the corresponding straight lines into a coordinate system.

$\begin{array}{rcl} (I) & 5 x & + & 3 y & = & 6 \\ (II) & - y & + & 7 & = & 0 \end{array}$
has exactly one solution
has no solution
has infinitely many solutions

For this task you need the following basic knowledge: System of linear equations
$(I)$ solve for $y$
$\begin{array}{rcll} 5 x + 3 y & = & 6 & | - 5 x \\ 3 y & = & - 5 x + 6 & | : 3 \\ y & = & - \frac{5}{3} x + 2 \end{array}$
$(II)$ solve for $y$
$\begin{array}{rcl} - y + 7 & = & 0 & | + y \\ y & = & 7 \end{array}$
Draw both straight lines in a coordinate system
Answer
The linear system of equations has exactly one solution, since the two straight lines intersect at the point $S$ .
First solve the equations $(I)$ and $(II)$ for y and draw the corresponding straight lines into a coordinate system.

$\begin{array}{rcl} (I) & - 3 x & + & 4 y & = & 10 \\ (II) & - x & + & 5 y & = & 17,5 \end{array}$
has exactly one solution
has no solutions
has infinitely many solutions

For this task you need the following basic knowledge: System of linear equations
$(I)$ solve for $y$
$\begin{array}{rcll} - 3 x + 4 y & = & 10 & | + 3 x \\ 4 y & = & 3 x + 10 & | : 4 \\ y & = & \frac{3}{4} x + 2,5 \end{array}$
$(II)$ solve for $y$
$\begin{array}{rcll} - x + 5 y & = & 17,5 & | + x \\ 5 y & = & x + 17,5 & | : 5 \\ y & = & \frac{1}{5} x + 3,5 \end{array}$
Draw both straight lines in a coordinate system
Answer
The linear system of equations has exactly one solution, since the two straight lines intersect at the point $S$ .
First solve equations $(I)$ and $(II)$ for y and draw the corresponding straight lines into a coordinate system.

$\begin{array}{rcl} (I) & x & + & 3 y & = & 9 \\ (II) & - 3 x & + & 4 y & = & 6 \end{array}$
has exactly one solution
no solution
has infinitely many solutions

For this task you need the following basic knowledge: System of linear equations
$(I)$ solve for $y$
$\begin{array}{rcll} x + 3 y & = & 9 & | - x \\ 3 y & = & - x + 9 & | : 3 \\ y & = & - \frac{1}{3} x + 3 \end{array}$
$(II)$ solve for $y$
$\begin{array}{rcll} - 3 x + 4 y & = & 6 & | + 3 x \\ 4 y & = & 3 x + 6 & | : 4 \\ y & = & \frac{3}{4} x + 1,5 \end{array}$
Draw both straight lines in a coordinate system
Answer
The linear system of equations has exactly one solution, since the two straight lines intersect at the point $S$ .
First solve equations $(I)$ and $(II)$ for y and draw the corresponding straight lines into a coordinate system.

Exercises: graphically solving systems of equations

Determining intersection points

Reading off intersection points

Determining intersection points

Reading off intersectio points

(I) solve for y

(II) solve for y

Answer

(I) solve for y

(II) solve for y

Answer

(I) solve for y

(II) solve for y

Answer

(I) solve for y

(II) solve for y

Answer

(I) solve for y

(II) solve for y

Answer

(I) solve for y

(II) solve for y

Answer

$(I)$ solve for $y$

$(II)$ solve for $y$

$(I)$ solve for $y$

$(II)$ solve for $y$

$(I)$ solve for $y$

$(II)$ solve for $y$

$(I)$ solve for $y$

$(II)$ solve for $y$

$(I)$ solve for $y$

$(II)$ solve for $y$

$(I)$ solve for $y$

$(II)$ solve for $y$