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Sagot :
To determine which equations represent a line that passes through the given points, we need to check each equation against all four points.
Let's break down each proposed equation and validate whether it satisfies all given points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -10 & 2 \\ \hline -4 & 1 \\ \hline 8 & -1 \\ \hline 14 & -2 \\ \hline \end{array} \][/tex]
### Equation 1: [tex]\( y - 2 = -6(x + 10) \)[/tex]
Convert this to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 2 = -6(x + 10) \implies y - 2 = -6x - 60 \implies y = -6x - 58 \][/tex]
Check if this equation passes through all the points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -6(-10) - 58 = 60 - 58 = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -6(-4) - 58 = 24 - 58 = -34 \][/tex] (Invalid)
So, Equation 1 does not pass through all points.
### Equation 2: [tex]\( y - 2 = -\frac{1}{6}(x + 10) \)[/tex]
Convert to slope-intercept form:
[tex]\[ y - 2 = -\frac{1}{6}(x + 10) \implies y - 2 = -\frac{1}{6}x - \frac{10}{6} \implies y = -\frac{1}{6}x - \frac{5}{3} + 6 \implies y = -\frac{1}{6}x + \frac{13}{3} \][/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + \frac{13}{3} = \frac{10}{6} + \frac{13}{3} = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-4) + \frac{13}{3} = \frac{4}{6} + \frac{13}{3} = 1 \][/tex] (Valid)
3. At [tex]\( (8, -1) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(8) + \frac{13}{3} = -\frac{8}{6} + \frac{13}{3} = -1 \][/tex] (Valid)
4. At [tex]\( (14, -2) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(14) + \frac{13}{3} = -\frac{14}{6} + \frac{13}{3} = -2 \][/tex] (Valid)
Equation 2 passes through all points.
### Equation 3: [tex]\( y - 1 = -\frac{1}{6}(x + 4) \)[/tex]
Convert to slope-intercept form:
[tex]\[ y - 1 = -\frac{1}{6}(x + 4) \implies y - 1 = -\frac{1}{6}x - \frac{4}{6} \implies y = -\frac{1}{6}x - \frac{2}{3} + 1 \implies y = -\frac{1}{6}x + \frac{1}{3} \][/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + \frac{1}{3} = \frac{10}{6} + \frac{1}{3} = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-4) + \frac{1}{3} = \frac{4}{6} + \frac{1}{3} = 1 \][/tex] (Valid)
3. At [tex]\( (8, -1) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(8) + \frac{1}{3} = -\frac{8}{6} + \frac{1}{3} = -1 \][/tex] (Valid)
4. At [tex]\( (14, -2) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(14) + \frac{1}{3} = -\frac{14}{6} + \frac{1}{3} = -2 \][/tex] (Valid)
Equation 3 passes through all points.
### Equation 4: [tex]\( y = -6x - 58 \)[/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -6(-10) - 58 = 60 - 58 = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -6(-4) - 58 = 24 - 58 = -34 \][/tex] (Invalid)
So, Equation 4 does not pass through all points.
### Equation 5: [tex]\( y = -\frac{1}{6}x + \frac{1}{3} \)[/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + \frac{1}{3} = \frac{10}{6} + \frac{1}{3} = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-4) + \frac{1}{3} = \frac{4}{6} + \frac{1}{3} = 1 \][/tex] (Valid)
3. At [tex]\( (8, -1) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(8) + \frac{1}{3} = -\frac{8}{6} + \frac{1}{3} = -1 \][/tex] (Valid)
4. At [tex]\( (14, -2) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(14) + \frac{1}{3} = -\frac{14}{6} + \frac{1}{3} = -2 \][/tex] (Valid)
Equation 5 passes through all points.
### Equation 6: [tex]\( y = -\frac{1}{6}x + 5 \)[/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + 5 = \frac{10}{6} + 5 = 2 \][/tex] (Invalid)
Equation 6 does not pass through all points.
Based on these checks, the valid equations are:
- Equation 2: [tex]\( y-2=-\frac{1}{6}(x+10) \)[/tex]
- Equation 3: [tex]\( y-1=-\frac{1}{6}(x+4) \)[/tex]
Thus, the equations that represent a line passing through the points given in the table are:
[tex]\[ \boxed{y - 2 = -\frac{1}{6}(x + 10) \quad \text{and} \quad y - 1 = -\frac{1}{6}(x + 4)} \][/tex]
Let's break down each proposed equation and validate whether it satisfies all given points:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline -10 & 2 \\ \hline -4 & 1 \\ \hline 8 & -1 \\ \hline 14 & -2 \\ \hline \end{array} \][/tex]
### Equation 1: [tex]\( y - 2 = -6(x + 10) \)[/tex]
Convert this to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 2 = -6(x + 10) \implies y - 2 = -6x - 60 \implies y = -6x - 58 \][/tex]
Check if this equation passes through all the points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -6(-10) - 58 = 60 - 58 = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -6(-4) - 58 = 24 - 58 = -34 \][/tex] (Invalid)
So, Equation 1 does not pass through all points.
### Equation 2: [tex]\( y - 2 = -\frac{1}{6}(x + 10) \)[/tex]
Convert to slope-intercept form:
[tex]\[ y - 2 = -\frac{1}{6}(x + 10) \implies y - 2 = -\frac{1}{6}x - \frac{10}{6} \implies y = -\frac{1}{6}x - \frac{5}{3} + 6 \implies y = -\frac{1}{6}x + \frac{13}{3} \][/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + \frac{13}{3} = \frac{10}{6} + \frac{13}{3} = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-4) + \frac{13}{3} = \frac{4}{6} + \frac{13}{3} = 1 \][/tex] (Valid)
3. At [tex]\( (8, -1) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(8) + \frac{13}{3} = -\frac{8}{6} + \frac{13}{3} = -1 \][/tex] (Valid)
4. At [tex]\( (14, -2) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(14) + \frac{13}{3} = -\frac{14}{6} + \frac{13}{3} = -2 \][/tex] (Valid)
Equation 2 passes through all points.
### Equation 3: [tex]\( y - 1 = -\frac{1}{6}(x + 4) \)[/tex]
Convert to slope-intercept form:
[tex]\[ y - 1 = -\frac{1}{6}(x + 4) \implies y - 1 = -\frac{1}{6}x - \frac{4}{6} \implies y = -\frac{1}{6}x - \frac{2}{3} + 1 \implies y = -\frac{1}{6}x + \frac{1}{3} \][/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + \frac{1}{3} = \frac{10}{6} + \frac{1}{3} = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-4) + \frac{1}{3} = \frac{4}{6} + \frac{1}{3} = 1 \][/tex] (Valid)
3. At [tex]\( (8, -1) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(8) + \frac{1}{3} = -\frac{8}{6} + \frac{1}{3} = -1 \][/tex] (Valid)
4. At [tex]\( (14, -2) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(14) + \frac{1}{3} = -\frac{14}{6} + \frac{1}{3} = -2 \][/tex] (Valid)
Equation 3 passes through all points.
### Equation 4: [tex]\( y = -6x - 58 \)[/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -6(-10) - 58 = 60 - 58 = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -6(-4) - 58 = 24 - 58 = -34 \][/tex] (Invalid)
So, Equation 4 does not pass through all points.
### Equation 5: [tex]\( y = -\frac{1}{6}x + \frac{1}{3} \)[/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + \frac{1}{3} = \frac{10}{6} + \frac{1}{3} = 2 \][/tex] (Valid)
2. At [tex]\((-4, 1)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-4) + \frac{1}{3} = \frac{4}{6} + \frac{1}{3} = 1 \][/tex] (Valid)
3. At [tex]\( (8, -1) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(8) + \frac{1}{3} = -\frac{8}{6} + \frac{1}{3} = -1 \][/tex] (Valid)
4. At [tex]\( (14, -2) \)[/tex]:
[tex]\[ y = -\frac{1}{6}(14) + \frac{1}{3} = -\frac{14}{6} + \frac{1}{3} = -2 \][/tex] (Valid)
Equation 5 passes through all points.
### Equation 6: [tex]\( y = -\frac{1}{6}x + 5 \)[/tex]
Check all points:
1. At [tex]\((-10, 2)\)[/tex]:
[tex]\[ y = -\frac{1}{6}(-10) + 5 = \frac{10}{6} + 5 = 2 \][/tex] (Invalid)
Equation 6 does not pass through all points.
Based on these checks, the valid equations are:
- Equation 2: [tex]\( y-2=-\frac{1}{6}(x+10) \)[/tex]
- Equation 3: [tex]\( y-1=-\frac{1}{6}(x+4) \)[/tex]
Thus, the equations that represent a line passing through the points given in the table are:
[tex]\[ \boxed{y - 2 = -\frac{1}{6}(x + 10) \quad \text{and} \quad y - 1 = -\frac{1}{6}(x + 4)} \][/tex]
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