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Chapter 2: Problem 43

Sketch the straight line defined by the linear equation by finding the \(x\) - and \(y\) -intercepts. $$ x+2 y-4=0 $$

Short Answer

Expert verified
The x-intercept of the linear equation \(x + 2y - 4 = 0\) is (4, 0) and the y-intercept is (0, 2). To sketch the straight line, connect these two points.

Step by step solution

01

Find the x-intercept

To find the x-intercept, we need to set y=0 in the equation and solve for x: \(x + 2(0) - 4 = 0\) \(x - 4 = 0\) \(x = 4\) So, the x-intercept is at the point (4, 0).
02

Find the y-intercept

To find the y-intercept, we need to set x=0 in the equation and solve for y: \(0 + 2y - 4 = 0\) \(2y - 4 = 0\) \(2y = 4\) \(y = 2\) So, the y-intercept is at the point (0, 2).
03

Sketch the straight line

Now that we have the x- and y-intercepts, we can sketch the straight line by connecting the points (4, 0) and (0, 2). You will see that the line passes through both of these intercepts, giving you a visual representation of the linear equation \(x + 2y - 4 = 0\).

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Key Concepts

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

X-Intercept
The x-intercept of a graph is where the line crosses the x-axis. This point has a y-coordinate of 0. To find the x-intercept of a linear equation, set y to 0 and solve for x. For instance, using the equation provided, \( x + 2y - 4 = 0 \), substitute \( y = 0 \). This simplifies to \( x - 4 = 0 \), leading us to \( x = 4 \). This means the line crosses the x-axis at the point (4, 0).
Understanding x-intercepts helps in graphing linear equations because it provides a starting or key point on the graph. You can visualize the x-intercept as the point where the graph touches or crosses the horizontal axis. This is crucial for understanding the behavior of linear equations on a graph.
Y-Intercept
The y-intercept is the point where the graph crosses the y-axis, with an x-coordinate of 0. To determine the y-intercept, set x to 0 in the equation and solve for y. Take our example, \( x + 2y - 4 = 0 \), and set \( x = 0 \). The equation becomes \( 2y - 4 = 0 \), which simplifies to \( y = 2 \). This reveals that the y-intercept is at the point (0, 2).
Knowing the y-intercept helps us in graphing because it provides another essential point of reference. It allows us to plot the line accurately on a coordinate plane by showing where the line intersects the vertical axis.
Determining both intercepts simplifies the process of sketching the equation on a graph and helps verify your work by seeing if both intercepts align correctly with the equation when drawn.
Graphing Linear Equations
Graphing linear equations involves plotting points that satisfy the equation and connecting them in a straight line. Once you have identified the x-intercept and y-intercept, drawing the line becomes straightforward. These intercepts are like anchors that help position the line accurately on the graph.
To graph the equation \( x + 2y - 4 = 0 \), plot the points (4, 0) and (0, 2) based on the intercepts calculated. Using a ruler, draw a straight line through these two points. This visually represents the equation on the coordinate plane.
  • The graph of a linear equation is always a straight line.
  • All points on the line are solutions to the equation.
  • Understanding the slope between these intercepts can provide more insight into the line鈥檚 steepness and direction.
Overall, graphing helps in visualizing the relationship between variables in a linear equation, offers a clear view of intercepts, and helps check solutions by comparing plotted points with the line.

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Most popular questions from this chapter

LINEAR DEPRECIATION Suppose an asset has an original value of \(\$ C\) and is depreciated linearly over \(N\) yr with a scrap value of \(\$ S\). Show that the asset's book value at the end of the \(t\) th year is described by the function $$ V(t)=C-\left(\frac{C-S}{N}\right) t $$ Hint: Find an equation of the straight line passing through the points \((0, C)\) and \((N, S)\). (Why?)

REVENUE OF PoLO RALPH LAUREN Citing strong sales and benefits from a new arm that will design lifestyle brands for department and specialty stores, the company projects revenue (in billions of dollars) to be $$ R(t)=-0.06 t^{2}+0.69 t+3.25 \quad(0 \leq t \leq 3) $$ in year \(t\), where \(t=0\) corresponds to 2005 . a. What was the revenue of the company in 2005 ? b. Find \(R(1), R(2)\), and \(R(3)\) and interpret your results.

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(b^{2}=4 a c\), then the graph of the quadratic function \(f(x)=\) \(a x^{2}+b x+c(a \neq 0)\) touches the \(x\) -axis at exactly one point.

DECISION ANALYSIS A product may be made using machine I or machine II. The manufacturer estimates that the monthly fixed costs of using machine I are \(\$ 18,000\), whereas the monthly fixed costs of using machine II are \(\$ 15,000\). The variable costs of manufacturing 1 unit of the product using machine I and machine II are \(\$ 15\) and \(\$ 20\), respectively. The product sells for \(\$ 50\) each. a. Find the cost functions associated with using each machine. b. Sketch the graphs of the cost functions of part (a) and the revenue functions on the same set of axes. c. Which machine should management choose in order to maximize their profit if the projected sales are 450 units? 550 units? 650 units? d. What is the profit for each case in part (c)?

PREVALENCE OF ALZHEIMER's PATIENTS Based on a study conducted in 1997 , the percent of the U.S. population by age afflicted with Alzheimer's disease is given by the function \(P(x)=0.0726 x^{2}+0.7902 x+4.9623 \quad(0 \leq x \leq 25)\) where \(x\) is measured in years, with \(x=0\) corresponding to age \(65 .\) What percent of the U.S. population at age 65 is expected to have Alzheimer's disease? At age 90 ?
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