Question

consider the system of equations shown:

$$left{\begin{array}{l}y = 6 - x\\y = -\frac{1}{2}x + 5end{array} ight.$$

a. circle a response.
the system has one solution / infinitely many solutions/no solution.
explain how you know.
b. solve the system of equations algebraically by using substitution.
if the system has one solution, write it as an ordered pair.
if the system has infinitely many solutions or no solution, represent the solution by using set - builder notation.

Explanation:

Step1: Substitute \( y = 6 - x \) into \( y = -\frac{1}{2}x + 5 \)

Since \( y=y \), we have \( 6 - x=-\frac{1}{2}x + 5 \)

Step2: Solve for \( x \)

Add \( x \) to both sides: \( 6=-\frac{1}{2}x + 5+x \), which simplifies to \( 6=\frac{1}{2}x + 5 \).
Subtract \( 5 \) from both sides: \( 6 - 5=\frac{1}{2}x \), so \( 1=\frac{1}{2}x \).
Multiply both sides by \( 2 \): \( x = 2 \)

Step3: Solve for \( y \)

Substitute \( x = 2 \) into \( y = 6 - x \). Then \( y=6 - 2=4 \)

Since we found a unique pair \( (x,y)=(2,4) \), the system has one solution.

Answer:

a. The system has one solution. We know this because when we solve the system algebraically (by substitution as shown above), we find a unique pair of values for \( x \) and \( y \).
b. The solution of the system as an ordered pair is \( (2,4) \)