Answer :
Certainly! To determine which equation among the given choices contains the coordinate pairs from the table, we need to check each equation against each point [tex]\((x, y)\)[/tex] in the table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 6 \\ \hline -2 & 3 \\ \hline -1 & 0 \\ \hline 0 & -3 \\ \hline 1 & -6 \\ \hline \end{array} \][/tex]
We will examine each point against the equations one by one:
1. Equation [tex]\( y = -3x - 1 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = -3(-3) - 1 = 9 - 1 = 8 \)[/tex] (Not matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -3(-2) - 1 = 6 - 1 = 5 \)[/tex] (Not matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = -3(-1) - 1 = 3 - 1 = 2 \)[/tex] (Not matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = -3(0) - 1 = 0 - 1 = -1 \)[/tex] (Not matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -3(1) - 1 = -3 - 1 = -4 \)[/tex] (Not matching [tex]\( y = -6 \)[/tex])
2. Equation [tex]\( y = 3x - 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = 3(-3) - 3 = -9 - 3 = -12 \)[/tex] (Not matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = 3(-2) - 3 = -6 - 3 = -9 \)[/tex] (Not matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = 3(-1) - 3 = -3 - 3 = -6 \)[/tex] (Not matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = 3(0) - 3 = 0 - 3 = -3 \)[/tex] (Matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = 3(1) - 3 = 3 - 3 = 0 \)[/tex] (Not matching [tex]\( y = -6 \)[/tex])
3. Equation [tex]\( y = -\frac{1}{3}x - 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = -\frac{1}{3}(-3) - 3 = 1 - 3 = -2 \)[/tex] (Not matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -\frac{1}{3}(-2) - 3 = \frac{2}{3} - 3 = -\frac{7}{3} \)[/tex] (Not matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = -\frac{1}{3}(-1) - 3 = \frac{1}{3} - 3 = -\frac{8}{3} \)[/tex] (Not matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = -\frac{1}{3}(0) - 3 = 0 - 3 = -3 \)[/tex] (Matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -\frac{1}{3}(1) - 3 = -\frac{1}{3} - 3 = -\frac{10}{3} \)[/tex] (Not matching [tex]\( y = -6 \)[/tex])
4. Equation [tex]\( y = -3x - 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = -3(-3) - 3 = 9 - 3 = 6 \)[/tex] (Matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -3(-2) - 3 = 6 - 3 = 3 \)[/tex] (Matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = -3(-1) - 3 = 3 - 3 = 0 \)[/tex] (Matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = -3(0) - 3 = 0 - 3 = -3 \)[/tex] (Matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -3(1) - 3 = -3 - 3 = -6 \)[/tex] (Matching [tex]\( y = -6 \)[/tex])
By testing each coordinate pair with the equations given, the only equation that matches all points in the table is:
[tex]\[ y = -3x - 3 \][/tex]
Thus, the correct equation that contains all the coordinate pairs is:
[tex]\[ y = -3x - 3 \][/tex]
The answer is: [tex]\( y = -3x - 3 \)[/tex], which corresponds to choice [tex]\( 4 \)[/tex].
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -3 & 6 \\ \hline -2 & 3 \\ \hline -1 & 0 \\ \hline 0 & -3 \\ \hline 1 & -6 \\ \hline \end{array} \][/tex]
We will examine each point against the equations one by one:
1. Equation [tex]\( y = -3x - 1 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = -3(-3) - 1 = 9 - 1 = 8 \)[/tex] (Not matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -3(-2) - 1 = 6 - 1 = 5 \)[/tex] (Not matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = -3(-1) - 1 = 3 - 1 = 2 \)[/tex] (Not matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = -3(0) - 1 = 0 - 1 = -1 \)[/tex] (Not matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -3(1) - 1 = -3 - 1 = -4 \)[/tex] (Not matching [tex]\( y = -6 \)[/tex])
2. Equation [tex]\( y = 3x - 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = 3(-3) - 3 = -9 - 3 = -12 \)[/tex] (Not matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = 3(-2) - 3 = -6 - 3 = -9 \)[/tex] (Not matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = 3(-1) - 3 = -3 - 3 = -6 \)[/tex] (Not matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = 3(0) - 3 = 0 - 3 = -3 \)[/tex] (Matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = 3(1) - 3 = 3 - 3 = 0 \)[/tex] (Not matching [tex]\( y = -6 \)[/tex])
3. Equation [tex]\( y = -\frac{1}{3}x - 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = -\frac{1}{3}(-3) - 3 = 1 - 3 = -2 \)[/tex] (Not matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -\frac{1}{3}(-2) - 3 = \frac{2}{3} - 3 = -\frac{7}{3} \)[/tex] (Not matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = -\frac{1}{3}(-1) - 3 = \frac{1}{3} - 3 = -\frac{8}{3} \)[/tex] (Not matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = -\frac{1}{3}(0) - 3 = 0 - 3 = -3 \)[/tex] (Matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -\frac{1}{3}(1) - 3 = -\frac{1}{3} - 3 = -\frac{10}{3} \)[/tex] (Not matching [tex]\( y = -6 \)[/tex])
4. Equation [tex]\( y = -3x - 3 \)[/tex]:
- For [tex]\( x = -3 \)[/tex]: [tex]\( y = -3(-3) - 3 = 9 - 3 = 6 \)[/tex] (Matching [tex]\( y = 6 \)[/tex])
- For [tex]\( x = -2 \)[/tex]: [tex]\( y = -3(-2) - 3 = 6 - 3 = 3 \)[/tex] (Matching [tex]\( y = 3 \)[/tex])
- For [tex]\( x = -1 \)[/tex]: [tex]\( y = -3(-1) - 3 = 3 - 3 = 0 \)[/tex] (Matching [tex]\( y = 0 \)[/tex])
- For [tex]\( x = 0 \)[/tex]: [tex]\( y = -3(0) - 3 = 0 - 3 = -3 \)[/tex] (Matching [tex]\( y = -3 \)[/tex])
- For [tex]\( x = 1 \)[/tex]: [tex]\( y = -3(1) - 3 = -3 - 3 = -6 \)[/tex] (Matching [tex]\( y = -6 \)[/tex])
By testing each coordinate pair with the equations given, the only equation that matches all points in the table is:
[tex]\[ y = -3x - 3 \][/tex]
Thus, the correct equation that contains all the coordinate pairs is:
[tex]\[ y = -3x - 3 \][/tex]
The answer is: [tex]\( y = -3x - 3 \)[/tex], which corresponds to choice [tex]\( 4 \)[/tex].
