Answer :
To determine which point lies on the line that goes through the point [tex]\((-4, -2)\)[/tex] and has an [tex]\(x\)[/tex]-intercept of [tex]\(-1\)[/tex], let's follow a step-by-step process.
First, let's find the slope of the line. The slope ([tex]\(m\)[/tex]) is calculated using the points [tex]\((-4, -2)\)[/tex] and [tex]\((-1, 0)\)[/tex] (the x-intercept).
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates:
[tex]\[ m = \frac{0 - (-2)}{-1 - (-4)} \][/tex]
[tex]\[ m = \frac{0 + 2}{-1 + 4} \][/tex]
[tex]\[ m = \frac{2}{3} \][/tex]
So, the slope of the line is [tex]\(\frac{2}{3}\)[/tex].
Next, let's find the y-intercept ([tex]\(b\)[/tex]) using the point [tex]\((-4, -2)\)[/tex] and the slope [tex]\(\frac{2}{3}\)[/tex]. The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substituting [tex]\(y = -2\)[/tex], [tex]\(x = -4\)[/tex], and [tex]\(m = \frac{2}{3}\)[/tex]:
[tex]\[ -2 = \frac{2}{3}(-4) + b \][/tex]
[tex]\[ -2 = -\frac{8}{3} + b \][/tex]
Now, solve for [tex]\(b\)[/tex]:
[tex]\[ b = -2 + \frac{8}{3} \][/tex]
[tex]\[ b = -\frac{6}{3} + \frac{8}{3} \][/tex]
[tex]\[ b = \frac{2}{3} \][/tex]
So, the y-intercept is [tex]\(\frac{2}{3}\)[/tex] and the equation of the line is:
[tex]\[ y = \frac{2}{3}x + \frac{2}{3} \][/tex]
Now, let's check which of the given points lies on this line by substituting each point into the equation.
1. For the point [tex]\((-6, -5)\)[/tex]:
[tex]\[ y = \frac{2}{3}(-6) + \frac{2}{3} \][/tex]
[tex]\[ y = -4 + \frac{2}{3} \][/tex]
[tex]\[ y = -4 + \frac{2}{3} = -\frac{12}{3} + \frac{2}{3} = -\frac{10}{3} \][/tex]
Since [tex]\(-5 \neq -\frac{10}{3}\)[/tex], the point [tex]\((-6, -5)\)[/tex] does not lie on the line.
2. For the point [tex]\((3, 2)\)[/tex]:
[tex]\[ y = \frac{2}{3}(3) + \frac{2}{3} \][/tex]
[tex]\[ y = 2 + \frac{2}{3} \][/tex]
[tex]\[ y = 2 + \frac{2}{3} = 2 + \frac{2}{3} \][/tex]
Hence, [tex]\(2 = 2 + \frac{2}{3}\)[/tex] is false, so the point [tex]\((3, 2)\)[/tex] is not on the line.
3. For the point [tex]\((4, 5)\)[/tex]:
[tex]\[ y = \frac{2}{3}(4) + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{8}{3} + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{10}{3} \][/tex]
Since [tex]\(5 \neq \frac{10}{3}\)[/tex], the point [tex]\((4, 5)\)[/tex] does not lie on the line.
4. For the point [tex]\((5, 4)\)[/tex]:
[tex]\[ y = \frac{2}{3}(5) + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{10}{3} + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{12}{3} \][/tex]
[tex]\[ y = 4 \][/tex]
Since [tex]\(4 \neq 4\)[/tex] is true, the point [tex]\((5, 4)\)[/tex] is not on the line.
Upon checking all points, none of the given points [tex]\((-6, -5)\)[/tex], [tex]\((3, 2)\)[/tex], [tex]\((4, 5)\)[/tex], or [tex]\((5, 4)\)[/tex] lies on the line described by the given conditions. Thus, the result is:
[tex]\[ None \ of \ the \ given \ points \ lies \ on \ the \ line. \][/tex]
First, let's find the slope of the line. The slope ([tex]\(m\)[/tex]) is calculated using the points [tex]\((-4, -2)\)[/tex] and [tex]\((-1, 0)\)[/tex] (the x-intercept).
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates:
[tex]\[ m = \frac{0 - (-2)}{-1 - (-4)} \][/tex]
[tex]\[ m = \frac{0 + 2}{-1 + 4} \][/tex]
[tex]\[ m = \frac{2}{3} \][/tex]
So, the slope of the line is [tex]\(\frac{2}{3}\)[/tex].
Next, let's find the y-intercept ([tex]\(b\)[/tex]) using the point [tex]\((-4, -2)\)[/tex] and the slope [tex]\(\frac{2}{3}\)[/tex]. The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
Substituting [tex]\(y = -2\)[/tex], [tex]\(x = -4\)[/tex], and [tex]\(m = \frac{2}{3}\)[/tex]:
[tex]\[ -2 = \frac{2}{3}(-4) + b \][/tex]
[tex]\[ -2 = -\frac{8}{3} + b \][/tex]
Now, solve for [tex]\(b\)[/tex]:
[tex]\[ b = -2 + \frac{8}{3} \][/tex]
[tex]\[ b = -\frac{6}{3} + \frac{8}{3} \][/tex]
[tex]\[ b = \frac{2}{3} \][/tex]
So, the y-intercept is [tex]\(\frac{2}{3}\)[/tex] and the equation of the line is:
[tex]\[ y = \frac{2}{3}x + \frac{2}{3} \][/tex]
Now, let's check which of the given points lies on this line by substituting each point into the equation.
1. For the point [tex]\((-6, -5)\)[/tex]:
[tex]\[ y = \frac{2}{3}(-6) + \frac{2}{3} \][/tex]
[tex]\[ y = -4 + \frac{2}{3} \][/tex]
[tex]\[ y = -4 + \frac{2}{3} = -\frac{12}{3} + \frac{2}{3} = -\frac{10}{3} \][/tex]
Since [tex]\(-5 \neq -\frac{10}{3}\)[/tex], the point [tex]\((-6, -5)\)[/tex] does not lie on the line.
2. For the point [tex]\((3, 2)\)[/tex]:
[tex]\[ y = \frac{2}{3}(3) + \frac{2}{3} \][/tex]
[tex]\[ y = 2 + \frac{2}{3} \][/tex]
[tex]\[ y = 2 + \frac{2}{3} = 2 + \frac{2}{3} \][/tex]
Hence, [tex]\(2 = 2 + \frac{2}{3}\)[/tex] is false, so the point [tex]\((3, 2)\)[/tex] is not on the line.
3. For the point [tex]\((4, 5)\)[/tex]:
[tex]\[ y = \frac{2}{3}(4) + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{8}{3} + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{10}{3} \][/tex]
Since [tex]\(5 \neq \frac{10}{3}\)[/tex], the point [tex]\((4, 5)\)[/tex] does not lie on the line.
4. For the point [tex]\((5, 4)\)[/tex]:
[tex]\[ y = \frac{2}{3}(5) + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{10}{3} + \frac{2}{3} \][/tex]
[tex]\[ y = \frac{12}{3} \][/tex]
[tex]\[ y = 4 \][/tex]
Since [tex]\(4 \neq 4\)[/tex] is true, the point [tex]\((5, 4)\)[/tex] is not on the line.
Upon checking all points, none of the given points [tex]\((-6, -5)\)[/tex], [tex]\((3, 2)\)[/tex], [tex]\((4, 5)\)[/tex], or [tex]\((5, 4)\)[/tex] lies on the line described by the given conditions. Thus, the result is:
[tex]\[ None \ of \ the \ given \ points \ lies \ on \ the \ line. \][/tex]
