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Chapter 5: Problem 2

Graph the equations. \(x+2 y=2\)

Short Answer

Expert verified
Plot (0,1) and (2,0), then draw a line through these points.

Step by step solution

01

- Identify the Equation's Form

The given equation is in the form of a linear equation: \(x + 2y = 2\). This can be rewritten in slope-intercept form \(y = mx + b\) to easily graph it.
02

- Rewrite in Slope-Intercept Form

Start by solving for \(y\). Subtract \(x\) from both sides:\[2y = -x + 2\]Then divide by 2:\[y = -\frac{1}{2}x + 1\]Now the equation is in slope-intercept form \(y = mx + b\), where \(m = -\frac{1}{2}\) is the slope and \(b = 1\) is the y-intercept.
03

- Identify the Y-Intercept

From the slope-intercept form equation \(y = -\frac{1}{2}x + 1\), we see that the y-intercept \(b\) is 1. This means the line crosses the y-axis at the point (0, 1).
04

- Use the Slope to Find Another Point

The slope \(-\frac{1}{2}\) means that for every 1 unit increase in \(x\), \(y\) decreases by \(\frac{1}{2}\) unit. Starting from the y-intercept (0, 1), if we move 2 units to the right (\(x\) increases by 2), then \(y\) will decrease by 1:Starting from (0, 1): New point is (2, 0).
05

- Plot the Points and Draw the Line

Plot the y-intercept (0, 1) and the point (2,0) on a graph. Draw a straight line through these points to represent the equation \(x + 2y = 2\).

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Key Concepts

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

linear equations
Linear equations are fundamental in algebra and are a type of equation where the highest power of the variable is 1. They can be written in various forms, but the most common ones are the standard form, ax + by = c, and the slope-intercept form, y = mx + b. Linear equations graph as straight lines on a coordinate plane. The main components of a linear equation include the coefficients (a, b, and c in standard form), the slope (m), and the y-intercept (b in slope-intercept form). Understanding these concepts will help you rearrange equations and graph them effectively.
Examples of linear equations include: x + 2y = 2 and y = -0.5x + 1.
slope-intercept form
The slope-intercept form is very useful for graphing linear equations. It's written as y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. This form makes it easy to identify the key features of the line. The slope 'm' shows how steep the line is, and the y-intercept 'b' indicates where the line crosses the y-axis. For instance, in the equation y = -0.5x + 1, the slope is -0.5, meaning the line goes down 0.5 units for every 1 unit it goes right. The y-intercept is 1, indicating the line crosses the y-axis at (0, 1).
  • To convert a standard form equation to slope-intercept form, isolate 'y' on one side of the equation.
  • This makes it easier to graph the equation because you can quickly identify slope and y-intercept.
graph plotting
Graph plotting involves plotting points on a coordinate plane to represent an equation. For linear equations, you'll need at least two points to plot a line. Start by finding the y-intercept, which is an easy point to identify from the slope-intercept form. Then use the slope to find another point. For example, with the equation y = -0.5x + 1, start at (0, 1). Then, using the slope -0.5, move 2 units to the right and 1 unit down to find another point (2,0).
  • Use graph paper or a digital tool to accurately plot your points.
  • Once you've plotted your points, draw a straight line through them to complete the graph.

Remember to label your axes and scale your graph appropriately.
y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. This is represented by 'b' in the slope-intercept form y = mx + b. To find the y-intercept from an equation, set x to 0 and solve for y. For example, in y = -0.5x + 1, when x = 0, y = 1, so the y-intercept is 1.
The y-intercept provides an easy starting point for graphing because it's an exact point on the graph. Once you have the y-intercept, you can use the slope to find more points. Just remember, the coordinate of the y-intercept is always in the form (0, b). Additionally, the y-intercept helps in understanding the behavior of the line in real-world applications, like initial values in economic models.
slope calculation
The slope of a line indicates its steepness and direction. It's calculated as the ratio of the change in y (rise) to the change in x (run). In mathematical terms, slope (m) is defined as \(\frac{rise}{run}\). For a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated as \(\frac{y_2 - y_1}{x_2 - x_1}\).
For instance, in the equation y = -0.5x + 1, the slope is -0.5. This means that for every 1 unit increase in x, y decreases by 0.5 units. Understanding the slope helps in graphing the line accurately.
  • Positive slopes mean the line ascends as you move right.
  • Negative slopes mean the line descends as you move right.
  • A slope of zero means the line is horizontal.
  • An undefined slope means the line is vertical.

Being able to calculate the slope will help you analyze and graph any linear equation quickly.

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Most popular questions from this chapter

The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration \(A_{0}\) of the drug in the bloodstream at the time of injection. However, the physician knows that after \(3 \mathrm{hr}\), the drug concentration in the blood is \(0.69 \mu \mathrm{g} / \mathrm{dL}\) and after \(4 \mathrm{hr}\), the concentration is \(0.655 \mu \mathrm{g} / \mathrm{dL}\). The model \(A(t)=A_{0} e^{-k t}\) represents the drug concentration \(A(t)\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) in the bloodstream \(t\) hours after injection. The value of \(k\) is a constant related to the rate at which the drug is removed by the body. a. Substitute 0.69 for \(A(t)\) and 3 for \(t\) in the model and write the resulting equation. b. Substitute 0.655 for \(A(t)\) and 4 for \(t\) in the model and write the resulting equation. c. Use the system of equations from parts (a) and (b) to solve for \(k .\) Round to 3 decimal places. d. Use the system of equations from parts (a) and (b) to approximate the initial concentration \(A_{0}\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) at the time of injection. Round to 2 decimal places. e. Determine the concentration of the drug after \(12 \mathrm{hr}\). Round to 2 decimal places.

a. Sketch the lines defined by \(y=x+2\) and \(y=-\frac{1}{2} x+2\) b. Find the area of the triangle bounded by the lines in part (a) and the \(x\) -axis.

To protect soil from erosion, some farmers plant winter cover crops such as winter wheat and rye. In addition to conserving soil, cover crops often increase crop yields in the row crops that follow in spring and summer. Suppose that a farmer has 800 acres of land and plans to plant winter wheat and rye. The input cost for 1 acre for each crop is given in the table along with the cost for machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Wheat } & \$ 90 & \$ 50 & \$ 42 \\ \hline \text { Rye } & \$ 120 & \$ 40 & \$ 35 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 90,000$$ for input costs and a maximum of $$\$ 36,000$$ for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre for wheat were $$\$ 40$$ and the profit per acre for rye were $$\$ 45$$, how many acres of each crop should be planted to maximize profit?

The population \(P(t)\) of a culture of bacteria grows exponentially for the first 72 hr according to the model \(P(t)=P_{0} e^{k t} .\) The variable \(t\) is the time in hours since the culture is started. The population of bacteria is 60,000 after 7 hr. The population grows to 80,000 after 12 hr. a. Determine the constant \(k\) to 3 decimal places. b. Determine the original population \(P_{0}\). Round to the nearest thousand. c. Determine the time required for the population to reach 300,000 . Round to the nearest hour.

Solve the system using any method. $$ \begin{array}{l} y=-\frac{1}{4} x+7 \\ y=-\frac{3}{2} x+17 \end{array} $$
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