IDNLearn.com makes it easy to find the right answers to your questions. Discover in-depth answers to your questions from our community of experienced professionals.
Sagot :
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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.
