Question

4-72. for each line below, make a table and graph. what do you notice? 4-72 hw etool homework help a. $y = \frac{2}{3}x - 1$ b. $2x - 3y = 3$

Explanation:

Part (a): \( y = \frac{2}{3}x - 1 \)

Step 1: Create a Table of Values

Choose \( x \)-values (e.g., \( x = -3, 0, 3 \)) and calculate corresponding \( y \)-values.

• For \( x = -3 \):

\( y = \frac{2}{3}(-3) - 1 = -2 - 1 = -3 \)

• For \( x = 0 \):

\( y = \frac{2}{3}(0) - 1 = 0 - 1 = -1 \)

• For \( x = 3 \):

\( y = \frac{2}{3}(3) - 1 = 2 - 1 = 1 \)

\( x \)	\( y \)
0	-1
3	1

Step 2: Graph the Line

Plot the points \((-3, -3)\), \((0, -1)\), and \((3, 1)\) on a coordinate plane and draw a straight line through them. The slope is \( \frac{2}{3} \) (rise 2, run 3) and the \( y \)-intercept is \(-1\).

Part (b): \( 2x - 3y = 3 \)

Step 1: Rewrite in Slope-Intercept Form (\( y = mx + b \))

Solve for \( y \):
\( 2x - 3y = 3 \)
Subtract \( 2x \) from both sides:
\( -3y = -2x + 3 \)
Divide by \(-3\):
\( y = \frac{2}{3}x - 1 \)

Step 2: Create a Table of Values (Same as Part (a))

Since the equation is identical to \( y = \frac{2}{3}x - 1 \), the table of values is the same:

\( x \)	\( y \)
0	-1
3	1

Step 3: Graph the Line

Plot the same points \((-3, -3)\), \((0, -1)\), \((3, 1)\) and draw a line—this is the same line as in Part (a).

What Do You Notice?

Both equations (\( y = \frac{2}{3}x - 1 \) and \( 2x - 3y = 3 \)) represent the same linear relationship. When simplified, \( 2x - 3y = 3 \) rearranges to \( y = \frac{2}{3}x - 1 \), so they have the same slope (\( \frac{2}{3} \)) and \( y \)-intercept (\(-1\)). Thus, their tables and graphs are identical.

Answer:

Part (a): \( y = \frac{2}{3}x - 1 \)

Step 1: Create a Table of Values

Choose \( x \)-values (e.g., \( x = -3, 0, 3 \)) and calculate corresponding \( y \)-values.

• For \( x = -3 \):

\( y = \frac{2}{3}(-3) - 1 = -2 - 1 = -3 \)

• For \( x = 0 \):

\( y = \frac{2}{3}(0) - 1 = 0 - 1 = -1 \)

• For \( x = 3 \):

\( y = \frac{2}{3}(3) - 1 = 2 - 1 = 1 \)

\( x \)	\( y \)
0	-1
3	1

Step 2: Graph the Line

Plot the points \((-3, -3)\), \((0, -1)\), and \((3, 1)\) on a coordinate plane and draw a straight line through them. The slope is \( \frac{2}{3} \) (rise 2, run 3) and the \( y \)-intercept is \(-1\).

Part (b): \( 2x - 3y = 3 \)

Step 1: Rewrite in Slope-Intercept Form (\( y = mx + b \))

Solve for \( y \):
\( 2x - 3y = 3 \)
Subtract \( 2x \) from both sides:
\( -3y = -2x + 3 \)
Divide by \(-3\):
\( y = \frac{2}{3}x - 1 \)

Step 2: Create a Table of Values (Same as Part (a))

Since the equation is identical to \( y = \frac{2}{3}x - 1 \), the table of values is the same:

\( x \)	\( y \)
0	-1
3	1

Step 3: Graph the Line

Plot the same points \((-3, -3)\), \((0, -1)\), \((3, 1)\) and draw a line—this is the same line as in Part (a).

What Do You Notice?

Both equations (\( y = \frac{2}{3}x - 1 \) and \( 2x - 3y = 3 \)) represent the same linear relationship. When simplified, \( 2x - 3y = 3 \) rearranges to \( y = \frac{2}{3}x - 1 \), so they have the same slope (\( \frac{2}{3} \)) and \( y \)-intercept (\(-1\)). Thus, their tables and graphs are identical.