Question

write an equation for each of the following: table with x values 5,7,9,11,13 and y values 13,17,21,25,29 graph of a line

Explanation:

Part 1: Equation from the Table

Step 1: Identify the pattern (slope)

We have \( x \) values: \( 5, 7, 9, 11, 13 \) and \( y \) values: \( 13, 17, 21, 25, 29 \).
The change in \( y \) (\( \Delta y \)) between consecutive \( x \) values: \( 17 - 13 = 4 \), \( 21 - 17 = 4 \), etc.
The change in \( x \) (\( \Delta x \)): \( 7 - 5 = 2 \), \( 9 - 7 = 2 \), etc.
Slope \( m = \frac{\Delta y}{\Delta x} = \frac{4}{2} = 2 \).

Step 2: Find the y-intercept (\( b \))

Use the point \( (x, y) = (5, 13) \) and \( y = mx + b \).
Substitute: \( 13 = 2(5) + b \)
\( 13 = 10 + b \)
Subtract 10: \( b = 3 \).

Step 3: Write the equation

Using \( y = mx + b \), with \( m = 2 \) and \( b = 3 \), we get \( y = 2x + 3 \).

Part 2: Equation from the Graph

Step 1: Identify two points

From the graph, the line passes through \( (0, 1) \) (y-intercept) and \( (4, 0) \) (x-intercept).

Step 2: Calculate the slope (\( m \))

Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 1}{4 - 0} = \frac{-1}{4} = -\frac{1}{4} \)? Wait, no—wait, let's recheck the graph. Wait, the line passes through \( (0, 1) \) and \( (4, 0) \)? Wait, no, looking at the axes: the y-axis is vertical (labeled \( y \)) and x-axis horizontal (labeled \( x \)). Wait, the line crosses the y-axis at \( (0, 1) \)? Wait, no, the graph: let's see the coordinates. Wait, the line goes through \( (0, 1) \) and \( (4, 0) \)? Wait, no, maybe I misread. Wait, the line passes through \( (0, 1) \) and \( (4, 0) \)? Wait, no, let's check the slope again. Wait, the line: when \( x = 0 \), \( y = 1 \); when \( x = 4 \), \( y = 0 \). So slope \( m = \frac{0 - 1}{4 - 0} = -\frac{1}{4} \)? Wait, no, maybe the points are \( (0, 1) \) and \( ( - 4, 2) \)? Wait, no, the graph: let's re-express. Wait, the line passes through \( (0, 1) \) and \( (4, 0) \)? Wait, no, maybe the correct points are \( (0, 1) \) and \( (4, 0) \), but let's check the slope again. Wait, no—wait, the line in the graph: let's see, the line goes through \( (0, 1) \) (y-intercept) and \( (4, 0) \) (x-intercept). So the slope \( m = \frac{0 - 1}{4 - 0} = -\frac{1}{4} \)? Wait, no, maybe I made a mistake. Wait, alternatively, the line passes through \( (0, 1) \) and \( ( - 4, 2) \)? No, let's do it properly.

Wait, the graph: the y-axis is vertical (up is positive \( y \)), x-axis horizontal (right is positive \( x \)). The line crosses the y-axis at \( (0, 1) \) (so \( b = 1 \)) and passes through \( (4, 0) \). So slope \( m = \frac{0 - 1}{4 - 0} = -\frac{1}{4} \). Wait, but let's check another point. If \( x = -4 \), then \( y = 2 \) (since from \( (0,1) \), moving left 4 (x=-4) and up 1 (y=2)). So slope \( m = \frac{2 - 1}{-4 - 0} = \frac{1}{-4} = -\frac{1}{4} \). So the equation is \( y = -\frac{1}{4}x + 1 \)? Wait, no—wait, maybe I misread the graph. Wait, the user’s graph: let's re-express. The line passes through \( (0, 1) \) and \( (4, 0) \), so the equation is \( y = -\frac{1}{4}x + 1 \)? Wait, no, let's check with \( x = 0 \), \( y = 1 \) (correct). \( x = 4 \), \( y = -\frac{1}{4}(4) + 1 = -1 + 1 = 0 \) (correct). So the equation is \( y = -\frac{1}{4}x + 1 \). Wait, but maybe the graph has different points. Alternatively, maybe the line passes through \( (0, 1) \) and \( ( - 4, 2) \), so slope \( m = \frac{2 - 1}{-4 - 0} = -\frac{1}{4} \), same as before.

Answer:

s:

• Table: \( \boldsymbol{y = 2x + 3} \)
• Graph: \( \boldsymbol{y = -\frac{1}{4}x + 1} \) (assuming the graph passes through \( (0, 1) \) and \( (4, 0) \))

(Note: If the graph’s points are different, adjust accordingly. For the table, the equation is definitely \( y = 2x + 3 \).)