A line has a gradient of [tex]$\frac{1}{3}$[/tex] and a [tex]$y$[/tex]-intercept of 2. Its equation will be
Answer (2)
The problem provides the gradient and y -intercept of a line.
We use the slope-intercept form of a linear equation: y = m x + c .
Substitute the given values m = 3 1 and c = 2 into the equation.
The equation of the line is y = 3 1 x + 2 .
Explanation
Understanding the Problem We are given that a line has a gradient of 3 1 and a y -intercept of 2. We need to find the equation of the line.
Using Slope-Intercept Form The slope-intercept form of a linear equation is given by y = m x + c , where m is the gradient (slope) and c is the y -intercept.
Substituting the Values We are given that the gradient m = 3 1 and the y -intercept c = 2 . Substituting these values into the slope-intercept form, we get: y = 3 1 x + 2
Finding the Equation Therefore, the equation of the line is y = 3 1 x + 2 .
Checking the Options The given option B is y = 2 1 x + 2 . Comparing this with our derived equation y = 3 1 x + 2 , we see that the gradient is different. Therefore, option B is incorrect.
Final Answer The correct equation of the line is y = 3 1 x + 2 .
Examples
Imagine you're climbing a hill. The gradient of the hill is like the slope of the line, telling you how steep it is. The y -intercept is where you start climbing on the y -axis. Knowing the gradient and y -intercept helps you map out your path, just like finding the equation of a line. This concept is useful in various real-life scenarios, such as designing ramps, calculating roof pitches, or even understanding the trajectory of a ball thrown in the air.
The equation of the line with a gradient of 3 1 and a y -intercept of 2 is y = 3 1 x + 2 . This format combines the slope and intercept into a clear linear equation. You can use this equation to find any point on the line or graph it accurately.
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