Question

consider the system of equations in standard form.
x + 4y = 26,
3x - 4y = 30
what is the best window range for the x - values to determine the solution?
what is the best window range for the y - values to determine the solution?
what is the exact solution to the system of equations?

Explanation:

For the system of equations \(

$$\begin{cases} x + 4y = 26 \\ 3x - 4y = 30 \end{cases}$$

\)

Step 1: Solve the system to find \( x \) and \( y \)

Add the two equations to eliminate \( y \):
\[

$$\begin{align*} (x + 4y) + (3x - 4y) &= 26 + 30 \\ 4x &= 56 \\ x &= 14 \end{align*}$$

\]
Substitute \( x = 14 \) into \( x + 4y = 26 \):
\[

$$\begin{align*} 14 + 4y &= 26 \\ 4y &= 12 \\ y &= 3 \end{align*}$$

\]

Step 2: Determine the best window range for \( x \)-values

The solution for \( x \) is \( 14 \). To see the solution clearly, a window range around \( 14 \) is suitable. A common range could be \( 0 \) to \( 20 \) (or a narrower range like \( 10 \) to \( 20 \)) to include \( x = 14 \).

Step 3: Determine the best window range for \( y \)-values

The solution for \( y \) is \( 3 \). A window range around \( 3 \) is suitable, such as \( 0 \) to \( 5 \) (or a narrower range like \( 2 \) to \( 4 \)) to include \( y = 3 \).

Step 4: Exact solution

The solution to the system is \( x = 14 \) and \( y = 3 \), so the solution is \( (14, 3) \).

Answer:

s:

• Best window range for \( x \)-values: A range that includes \( 14 \), e.g., \( 0 \) to \( 20 \) (or similar, depending on the available options).
• Best window range for \( y \)-values: A range that includes \( 3 \), e.g., \( 0 \) to \( 5 \) (or similar, depending on the available options).
• Exact solution: \( (14, 3) \)