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Chapter 1: Problem 83

Find an equation of the line, in slope-intercept form, having the given properties. \(x\) -intercept: \(\left(\frac{1}{2}, 0\right) ; y\) -intercept: (0,3)

Short Answer

Expert verified
The Equation of the line in slope-intercept form is y = -6x + 3

Step by step solution

01

Identify the coordinates of the intercept points

From the problem, the x-intercept, which is the point where the line crosses the x-axis, is (1/2, 0) and the y-intercept, which is the point where the line crosses the y-axis, is (0, 3). So x1 = 1/2, y1 = 0, x2 = 0, y2 = 3
02

Calculate the Slope

The slope of the line (m) can be calculated using the formula, m = (y2 - y1) / (x2 - x1). Substituting the identified coordinates into the formula: m = (3 - 0) / (0 - 1/2) = -6
03

Write the equation in slope-intercept form

The slope-intercept form of a line is 'y = mx + b'. We know m (slope) = -6 and b (the y-intercept) = 3. Substituting m and b into the equation gives the final equation of the line as y = -6x + 3

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
The slope tells us how steep a line is. To find this steepness, we need two points on the line. In our case, they are the x-intercept and y-intercept.

We use the formula:
  • \( m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \)
This formula helps us determine the change in y relative to the change in x, giving us the line's slope \( m \).

Let's substitute the coordinates we have:
  • \( x_1 = \frac{1}{2} \), \( y_1 = 0 \) \( (x\text{-intercept}) \)
  • \( x_2 = 0 \), \( y_2 = 3 \) \( (y\text{-intercept}) \)
Using these,
  • \( m = \frac{3 - 0}{0 - \frac{1}{2}} \)
  • Simplifying, \( m = \frac{3}{-\frac{1}{2}} = -6 \)
So, the slope of the line is \(-6\). This means for every unit the x-value increases, the y-value decreases by 6 units.
Y-intercept
The y-intercept is an important feature of a line in slope-intercept form. It is the point where the line crosses the y-axis. This point is often represented by the letter \( b \) in the equation of a line: \( y = mx + b \).

In our exercise, the y-intercept is given as \( (0, 3) \). This means when \( x = 0 \), \( y = 3 \). Therefore, the y-intercept \( b = 3 \).

Knowing the y-intercept helps to quickly sketch the line on a graph as it gives you a starting point for drawing the line based on its slope. The equation representing the line can be written straight away as:
  • \( y = -6x + 3 \)
This equation merges the slope we calculated and the y-intercept found directly from the problem.
X-intercept
The x-intercept is the point where the line crosses the x-axis. This is crucial because it gives us one of the anchor points needed to find out more about the line, such as its equation.

An x-intercept is a point \( (a, 0) \) where the y-value is zero. From the problem, our x-intercept is \( \left( \frac{1}{2}, 0 \right) \).

To understand how the x-intercept relates to the line equation, remember that in the equation \( y = mx + b \), when \( y = 0 \), \( x \) must take a specific value:
  • \( 0 = -6x + 3 \)
  • Solving for \( x \), we subtract \( 3 \) from both sides: \( -3 = -6x \)
  • Divide by \( -6 \): \( x = \frac{1}{2} \)
This matches our initial x-intercept from the problem, confirming the line passes through this point on the graph.

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