Use the table to answer the question.

Question

x-value y-value if y=2x−5
y-value if y=x−3

0 −5
−3

1 −3
−2

2 −1
−1

3 1 0
Jian made a table showing some x- and y-values in a system of equations. Use his table to identify the solution for the system.

(1 point)
Responses

(2,−1)
left parenthesis 2 comma negative 1 right parenthesis

There is no solution.
There is no solution.

There is an infinite number of solutions.
There is an infinite number of solutions.

(3,0)

Answer 1

To find the solution for the system of equations using Jian's table, we can look for the pair of x- and y-values where the outputs from both equations are the same.

From the table we have:

For the equation \(y = 2x - 5\):

• At \(x = 0\), \(y = -5\)
• At \(x = 1\), \(y = -3\)
• At \(x = 2\), \(y = -1\)
• At \(x = 3\), \(y = 1\)

For the equation \(y = x - 3\):

• At \(x = 0\), \(y = -3\)
• At \(x = 1\), \(y = -2\)
• At \(x = 2\), \(y = -1\)
• At \(x = 3\), \(y = 0\)

Now, let's look for the x-value where the y-values from both equations are equal:

1. At \(x = 0\):

   • \(y = 2(0) - 5 = -5\)
   • \(y = 0 - 3 = -3\)

2. At \(x = 1\):

   • \(y = 2(1) - 5 = -3\)
   • \(y = 1 - 3 = -2\)

3. At \(x = 2\):

   • \(y = 2(2) - 5 = -1\)
   • \(y = 2 - 3 = -1\) (same value)

4. At \(x = 3\):

   • \(y = 2(3) - 5 = 1\)
   • \(y = 3 - 3 = 0\)

The only point where the y-values from both equations are equal is \( (2, -1) \).

Thus, the solution to the system of equations is:

(2, −1).