Question

y = \frac{5}{3}x + 1
answers must be written as fractions in simplest form, but you only need to complete 3 rows (can skip the rest). hint: try to find rows that would not result in fractions.
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$$\begin{tabular}{|c|c|c|} \\hline $x$ & $\\frac{5}{3}x$ & $y = \\frac{5}{3}x + 1$ \\\\ \\hline $-3$ & \\fbox{} & \\fbox{} \\\\ \\hline $-2$ & & \\\\ \\hline $-1$ & & \\\\ \\hline $0$ & & \\\\ \\hline $1$ & & \\\\ \\hline $2$ & & \\\\ \\hline $3$ & & \\\\ \\hline \\end{tabular}$$

Explanation:

To complete the table for the equation \( y = \frac{5}{3}x + 1 \), we substitute each value of \( x \) into the expressions \( \frac{5}{3}x \) and \( y = \frac{5}{3}x + 1 \). We'll focus on the row where \( x = -3 \) first, then show a few more for clarity.

Step 1: For \( x = -3 \)

• Calculate \( \frac{5}{3}x \):

Substitute \( x = -3 \) into \( \frac{5}{3}x \):

$$ \frac{5}{3}(-3) = \frac{5 \cdot (-3)}{3} = -5 $$

• Calculate \( y = \frac{5}{3}x + 1 \):

Substitute \( \frac{5}{3}x = -5 \) (from above) into the equation:

$$ y = -5 + 1 = -4 $$

Step 2: For \( x = -2 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(-2) = -\frac{10}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = -\frac{10}{3} + 1 = -\frac{10}{3} + \frac{3}{3} = -\frac{7}{3} $$

Step 3: For \( x = -1 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(-1) = -\frac{5}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = -\frac{5}{3} + 1 = -\frac{5}{3} + \frac{3}{3} = -\frac{2}{3} $$

Step 4: For \( x = 0 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(0) = 0 $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = 0 + 1 = 1 $$

Step 5: For \( x = 1 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(1) = \frac{5}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = \frac{5}{3} + 1 = \frac{5}{3} + \frac{3}{3} = \frac{8}{3} $$

Step 6: For \( x = 2 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(2) = \frac{10}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = \frac{10}{3} + 1 = \frac{10}{3} + \frac{3}{3} = \frac{13}{3} $$

Step 7: For \( x = 3 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(3) = 5 $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = 5 + 1 = 6 $$

Filled Table (Key Rows):

\( x \)	\( \frac{5}{3}x \)	\( y = \frac{5}{3}x + 1 \)
\( -2 \)	\( -\frac{10}{3} \)	\( -\frac{7}{3} \)
\( -1 \)	\( -\frac{5}{3} \)	\( -\frac{2}{3} \)
\( 0 \)	\( 0 \)	\( 1 \)
\( 1 \)	\( \frac{5}{3} \)	\( \frac{8}{3} \)
\( 2 \)	\( \frac{10}{3} \)	\( \frac{13}{3} \)
\( 3 \)	\( 5 \)	\( 6 \)

For the highlighted row (\( x = -3 \)):

• \( \frac{5}{3}x = \boldsymbol{-5} \)
• \( y = \frac{5}{3}x + 1 = \boldsymbol{-4} \)

Answer:

To complete the table for the equation \( y = \frac{5}{3}x + 1 \), we substitute each value of \( x \) into the expressions \( \frac{5}{3}x \) and \( y = \frac{5}{3}x + 1 \). We'll focus on the row where \( x = -3 \) first, then show a few more for clarity.

Step 1: For \( x = -3 \)

• Calculate \( \frac{5}{3}x \):

Substitute \( x = -3 \) into \( \frac{5}{3}x \):

$$ \frac{5}{3}(-3) = \frac{5 \cdot (-3)}{3} = -5 $$

• Calculate \( y = \frac{5}{3}x + 1 \):

Substitute \( \frac{5}{3}x = -5 \) (from above) into the equation:

$$ y = -5 + 1 = -4 $$

Step 2: For \( x = -2 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(-2) = -\frac{10}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = -\frac{10}{3} + 1 = -\frac{10}{3} + \frac{3}{3} = -\frac{7}{3} $$

Step 3: For \( x = -1 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(-1) = -\frac{5}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = -\frac{5}{3} + 1 = -\frac{5}{3} + \frac{3}{3} = -\frac{2}{3} $$

Step 4: For \( x = 0 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(0) = 0 $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = 0 + 1 = 1 $$

Step 5: For \( x = 1 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(1) = \frac{5}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = \frac{5}{3} + 1 = \frac{5}{3} + \frac{3}{3} = \frac{8}{3} $$

Step 6: For \( x = 2 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(2) = \frac{10}{3} $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = \frac{10}{3} + 1 = \frac{10}{3} + \frac{3}{3} = \frac{13}{3} $$

Step 7: For \( x = 3 \)

• Calculate \( \frac{5}{3}x \):

$$ \frac{5}{3}(3) = 5 $$

• Calculate \( y = \frac{5}{3}x + 1 \):

$$ y = 5 + 1 = 6 $$

Filled Table (Key Rows):

\( x \)	\( \frac{5}{3}x \)	\( y = \frac{5}{3}x + 1 \)
\( -2 \)	\( -\frac{10}{3} \)	\( -\frac{7}{3} \)
\( -1 \)	\( -\frac{5}{3} \)	\( -\frac{2}{3} \)
\( 0 \)	\( 0 \)	\( 1 \)
\( 1 \)	\( \frac{5}{3} \)	\( \frac{8}{3} \)
\( 2 \)	\( \frac{10}{3} \)	\( \frac{13}{3} \)
\( 3 \)	\( 5 \)	\( 6 \)

For the highlighted row (\( x = -3 \)):

• \( \frac{5}{3}x = \boldsymbol{-5} \)
• \( y = \frac{5}{3}x + 1 = \boldsymbol{-4} \)