Determine The Value Of \[$ X \$\] Given The Points \[$(x, 2)\$\] And \[$(6, 3)\$\] With A Slope \[$ M = -\frac{1}{2} \$\].
Determine the Value of { x $}$ Given the Points {(x, 2)$}$ and {(6, 3)$}$ with a Slope { m = -\frac{1}{2} $}$
In mathematics, the slope-intercept form of a linear equation is given by { y = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept. Given two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, we can determine the slope of the line passing through these points using the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$. In this article, we will use this formula to determine the value of { x $}$ given the points { (x, 2) $}$ and { (6, 3) $}$ with a slope { m = -\frac{1}{2} $}$.
The Slope-Intercept Form of a Linear Equation
The slope-intercept form of a linear equation is given by { y = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept. The slope { m $}$ can be determined using the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$, where { (x_1, y_1) $}$ and { (x_2, y_2) $}$ are two points on the line.
Determining the Slope
Given the points { (x, 2) $}$ and { (6, 3) $}$, we can determine the slope of the line passing through these points using the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$. Substituting the values of the points into the formula, we get:
{ m = \frac{3 - 2}{6 - x} $}$
Simplifying the expression, we get:
{ m = \frac{1}{6 - x} $}$
However, we are given that the slope { m $}$ is equal to { -\frac{1}{2} $}$. Therefore, we can set up the equation:
{ -\frac{1}{2} = \frac{1}{6 - x} $}$
Solving for { x $}$
To solve for { x $}$, we can start by cross-multiplying the equation:
{ -\frac{1}{2} (6 - x) = 1 $}$
Expanding the left-hand side of the equation, we get:
{ -3 + \frac{x}{2} = 1 $}$
Adding 3 to both sides of the equation, we get:
{ \frac{x}{2} = 4 $}$
Multiplying both sides of the equation by 2, we get:
{ x = 8 $}$
Therefore, the value of { x $}$ is 8.
In this article, we used the slope-intercept form of a linear equation to determine the value of { x $}$ given the points { (x, 2) $}$ and { (6, 3) $}$ with a slope { m = -\frac{1}{2} $}$. We first determined the slope of the line passing through the points using the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$. We then set up an equation using the given slope and solved for { x $}$. The final answer was { x = 8 $}$.
Here are a few additional examples of how to use the slope-intercept form of a linear equation to determine the value of { x $}$:
- Given the points { (x, 4) $}$ and { (2, 6) $}$ with a slope { m = \frac{1}{3} $}$, determine the value of { x $}$.
- Given the points { (x, 1) $}$ and { (5, 3) $}$ with a slope { m = -\frac{1}{4} $}$, determine the value of { x $}$.
- Given the points { (x, 2) $}$ and { (3, 4) $}$ with a slope { m = \frac{2}{3} $}$, determine the value of { x $}$.
Q: What is the slope-intercept form of a linear equation?
A: The slope-intercept form of a linear equation is given by { y = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept.
Q: How do I determine the slope of a line passing through two points?
A: To determine the slope of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$.
Q: What is the formula for determining the slope of a line passing through two points?
A: The formula for determining the slope of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$ is { m = \frac{y_2 - y_1}{x_2 - x_1} $}$.
Q: How do I determine the value of { x $}$ given two points and a slope?
A: To determine the value of { x $}$ given two points and a slope, you can use the slope-intercept form of a linear equation { y = mx + b $}$ and solve for { x $}$.
Q: What is the y-intercept in the slope-intercept form of a linear equation?
A: The y-intercept in the slope-intercept form of a linear equation { y = mx + b $}$ is the value of { b $}$, which is the point where the line intersects the y-axis.
Q: How do I find the y-intercept of a line passing through two points?
A: To find the y-intercept of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the formula { b = y_1 - m(x_1) $}$.
Q: What is the significance of the slope in the slope-intercept form of a linear equation?
A: The slope in the slope-intercept form of a linear equation { y = mx + b $}$ represents the rate of change of the line. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.
Q: How do I determine the equation of a line passing through two points?
A: To determine the equation of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the slope-intercept form of a linear equation { y = mx + b $}$ and solve for { b $}$.
Q: What is the equation of a line passing through two points?
A: The equation of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$ is given by { y = mx + b $}$, where { m $}$ is the slope and { b $}$ is the y-intercept.
Q: How do I graph a line passing through two points?
A: To graph a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the slope-intercept form of a linear equation { y = mx + b $}$ and plot the points on a coordinate plane.
Q: What is the significance of the x-intercept in the slope-intercept form of a linear equation?
A: The x-intercept in the slope-intercept form of a linear equation { y = mx + b $}$ represents the point where the line intersects the x-axis.
Q: How do I find the x-intercept of a line passing through two points?
A: To find the x-intercept of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the formula { x = \frac{-b}{m} $}$.
Q: What is the equation of a line passing through two points and a slope?
A: The equation of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$ and a slope { m $}$ is given by { y = mx + b $}$, where { b $}$ is the y-intercept.
Q: How do I determine the value of { x $}$ given two points and a slope?
A: To determine the value of { x $}$ given two points and a slope, you can use the slope-intercept form of a linear equation { y = mx + b $}$ and solve for { x $}$.
Q: What is the significance of the slope in the slope-intercept form of a linear equation?
A: The slope in the slope-intercept form of a linear equation { y = mx + b $}$ represents the rate of change of the line. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.
Q: How do I find the slope of a line passing through two points?
A: To find the slope of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$.
Q: What is the equation of a line passing through two points and a slope?
A: The equation of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$ and a slope { m $}$ is given by { y = mx + b $}$, where { b $}$ is the y-intercept.
Q: How do I determine the value of { x $}$ given two points and a slope?
A: To determine the value of { x $}$ given two points and a slope, you can use the slope-intercept form of a linear equation { y = mx + b $}$ and solve for { x $}$.
Q: What is the significance of the y-intercept in the slope-intercept form of a linear equation?
A: The y-intercept in the slope-intercept form of a linear equation { y = mx + b $}$ represents the point where the line intersects the y-axis.
Q: How do I find the y-intercept of a line passing through two points?
A: To find the y-intercept of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the formula { b = y_1 - m(x_1) $}$.
Q: What is the equation of a line passing through two points and a slope?
A: The equation of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$ and a slope { m $}$ is given by { y = mx + b $}$, where { b $}$ is the y-intercept.
Q: How do I determine the value of { x $}$ given two points and a slope?
A: To determine the value of { x $}$ given two points and a slope, you can use the slope-intercept form of a linear equation { y = mx + b $}$ and solve for { x $}$.
Q: What is the significance of the slope in the slope-intercept form of a linear equation?
A: The slope in the slope-intercept form of a linear equation { y = mx + b $}$ represents the rate of change of the line. A positive slope indicates that the line is increasing, while a negative slope indicates that the line is decreasing.
Q: How do I find the slope of a line passing through two points?
A: To find the slope of a line passing through two points { (x_1, y_1) $}$ and { (x_2, y_2) $}$, you can use the formula { m = \frac{y_2 - y_1}{x_2 - x_1} $}$.
Q: What is the equation of a line passing through two points and a slope?
A: The equation of a line passing through two points [$ (x_1