Question

c) write a system of linear equations that has the solution (0, -9).

Explanation:

Step1: Recall linear equation form

A linear equation is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For a system of linear equations with solution \((0, - 9)\), we can create two linear equations.

Step2: Create first equation

Let's take a simple case where the slope \( m = 1 \). Using the point - slope form or the slope - intercept form. Since the y - intercept \( b\) when \( x = 0\) is \( y=-9\). For the first equation, let's use \( y=x + b\). Substitute \( x = 0\) and \( y=-9\) into \( y=x + b\), we get \(-9=0 + b\), so \( b=-9\). The first equation is \( y=x - 9\).

Step3: Create second equation

Let's take the slope \( m = 2 \). Using the slope - intercept form \( y=mx + b\). Substitute \( x = 0\) and \( y=-9\) into \( y = 2x + b\), we have \(-9=2\times0 + b\), so \( b=-9\). The second equation is \( y = 2x-9\).

We can also use a simpler approach. For example, the equation \( x=0\) (a vertical line) and \( y=-9\) (a horizontal line) also form a system of linear equations with solution \((0,-9)\). Let's verify:

• For the system \(

$$\begin{cases}x = 0\\y=-9\end{cases}$$

\), when \( x = 0\) and \( y=-9\), both equations are satisfied.

• For the system \(

$$\begin{cases}y=x - 9\\y=2x-9\end{cases}$$

\), substitute \( x = 0\) into the first equation: \( y=0 - 9=-9\). Substitute \( x = 0\) into the second equation: \( y=2\times0-9=-9\).

Answer:

One possible system of linear equations is \(\boldsymbol{

$$\begin{cases}x = 0\\y=-9\end{cases}$$

}\) or \(\boldsymbol{

$$\begin{cases}y=x - 9\\y=2x-9\end{cases}$$

}\) (There are infinitely many correct systems, this is just an example).