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Chapter 2: Problem 90

Find an equation of the line whose x-intercept is 2 and y-intercept 3 .

Short Answer

Expert verified
The equation of the line whose x-intercept is 2 and y-intercept is 3 is \(y = -1.5x + 3\)

Step by step solution

01

Identify the intercepts

The x-intercept and y-intercept are given, (2,0) and (0,3) respectively. These are the points where the line crosses the x and y-axis.
02

Calculate the slope of the line

The equation to calculate the slope (m) is \[m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\]. Using the intercepts as two points on the line, plug these values into the slope formula to get \[m = \frac{(3 - 0)}{(0 - 2)} = -1.5\]. This is the slope of the line.
03

Write the line's equation

Now that we have the y-intercept (b=3) and the slope (m=-1.5), we can write the equation of the line in the form y = mx + b, which becomes \(y = -1.5x + 3\)

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Key Concepts

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

X-Intercept
In the realm of coordinate geometry, the x-intercept of a line is the point where the line crosses the x-axis. It has particular significance as it represents the value of the 'x' coordinate when the 'y' coordinate is zero. To find the x-intercept from an equation, simply set the 'y' value to zero and solve for 'x'.

In the provided exercise, the x-intercept is given as 2. This means that the line crosses the x-axis at the point (2, 0). This is a critical piece of information because, along with the y-intercept, it allows us to determine the slope of the line and subsequently the entire equation of the line.
Y-Intercept
The y-intercept is another foundational concept in understanding linear equations. It is defined as the point at which the line crosses the y-axis. Analogous to the x-intercept, it represents the 'y' coordinate when 'x' is zero. To locate the y-intercept on an equation, we set 'x' to zero and solve for 'y'.

The exercise tells us that the y-intercept is 3, indicating the line meets the y-axis at (0, 3). Knowing both intercepts is advantageous because it gives us two specific points on the line, which is exactly what we need to compute the slope and express the whole equation of the line.
Slope Calculation
The slope is a measure of how steep a line is and is denoted by 'm'. The slope of a line passing through two points, say (x1, y1) and (x2, y2), is computed using the slope formula \(m = \dfrac{(y2 - y1)}{(x2 - x1)}\). It represents the change in 'y' over the change in 'x'.

In our step-by-step solution, we calculated the slope using the x-intercept (2,0) and y-intercept (0,3), yielding a slope of -1.5. This number tells us that for every unit we move horizontally to the right, we move 1.5 units down vertically. Slope is a fundamental component to any linear equation because it defines the line's angle relative to the axes.
Linear Equations
A linear equation represents a straight line on a two-dimensional graph. The standard form of a linear equation is \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. This equation showcases that for any given x-value, the corresponding y-value can be found using the slope and the y-intercept.

Applying the concept to our exercise, we used the slope of -1.5 and the y-intercept of 3 to construct the line's equation: \(y = -1.5x + 3\). This concise formula allows us to understand and predict the behavior of the line across the entire graph, providing us with the capability to calculate the 'y' value for any 'x' value on this specific line.

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Most popular questions from this chapter

Assume a linear relationship holds. A person who weighs 150 pounds has 60 pounds of muscles, and a person that weighs 180 pounds has 72 pounds of muscles. If \(x\) represents the body weight and \(y\) the muscle weight, write an equation describing their relationship. Use this relationship to determine the muscle weight of a person that weighs 170 pounds.

Write an equation of the line satisfying the following conditions. Write the equation in the form \(\mathrm{Ax}+\mathrm{By}=C\). It passes through (-2,1) and \(m=-3 / 2\).

Graph the following three equations on the same set of coordinate axes. $$y=x+1 y=2 x+1 y=-x+1$$

Find the slope of the line whose equation is \(y=-3 x+5\).

Graph the line that passes through the given point and has the given slope. (2,3) and \(m=0\)
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