Question

identify the system of linear equations from the tables of values given below.

table 1:
\\(\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 0 & 5 \\\\ \\hline 5 & 0 \\\\ \\hline 2 & 3 \\\\ \\hline 4 & 1 \\\\ \\hline \\end{array}$$

\\)

table 2:
\\(\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline 0 & -3 \\\\ \\hline 3 & 0 \\\\ \\hline 5 & 2 \\\\ \\hline -1 & -4 \\\\ \\hline \\end{array}$$

\\)

options:

\\(\

$$\begin{aligned} y &= 2x - 5 \\\\ y &= 2x + 3 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} y &= -2x + 5 \\\\ y &= 2x - 3 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} y &= x - 5 \\\\ y &= -x + 3 \\end{aligned}$$

\\)

\\(\

$$\begin{aligned} y &= -x + 5 \\\\ y &= x - 3 \\end{aligned}$$

\\)

Explanation:

🆕 New Concept Discovered: Writing linear equations from tables
Finding the slope and y-intercept from given points

Step 1: Find the first equation

Using the first table:

• When \( x = 0 \), \( y = 5 \). This gives the y-intercept:

$$ b = 5 $$

• Use another point, such as \( (5, 0) \), to find the slope \( m \):

$$ m = \frac{0 - 5}{5 - 0} = \frac{-5}{5} = -1 $$

• Write the equation in slope-intercept form \( y = mx + b \):

$$ y = -x + 5 $$

Step 2: Find the second equation

Using the second table:

• When \( x = 0 \), \( y = -3 \). This gives the y-intercept:

$$ b = -3 $$

• Use another point, such as \( (3, 0) \), to find the slope \( m \):

$$ m = \frac{0 - (-3)}{3 - 0} = \frac{3}{3} = 1 $$

• Write the equation in slope-intercept form \( y = mx + b \):

$$ y = x - 3 $$

Step 3: Match with the options

The system of equations is:

$$
LATEXBLOCK0
$$

This matches the fourth option.

Answer:

$$
LATEXBLOCK0
$$