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Chapter 5: Problem 79

Given \(f(x)=m x+b,\) find \(m\) and \(b\) if \(f(3)=-3\) and \(f(-12)=-8\).

Short Answer

Expert verified
m = \frac{1}{3}, b = -4

Step by step solution

01

- Set up the equations using given points

To find the constants, use the given points to create equations. Plug in the first point \(3, -3\) into the function \(f(x) = mx + b\). This gives: \[-3 = 3m + b\]
02

- Plug in the second point

Now, plug in the second point \(-12, -8\) into the function \(f(x) = mx + b\). This gives: \[-8 = -12m + b\]
03

- Solve the system of equations

We have two equations: \[-3 = 3m + b\] and \[-8 = -12m + b\]. Solve this system of equations by elimination or substitution. Subtract the first equation from the second: \[-8 - (-3) = -12m - 3m \rightarrow -5 = -15m \rightarrow m = \frac{1}{3}\]
04

- Find the value of b

Substitute \(m = \frac{1}{3}\) back into the first equation: \[-3 = 3(\frac{1}{3}) + b \rightarrow -3 = 1 + b \rightarrow b = -4\]

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Key Concepts

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

solving systems of equations
When you encounter a system of equations, it means you're working with two or more equations that share the same set of variables. In this case, we have two equations that need to be solved simultaneously to find the values of the constants. Here's how to solve them step-by-step:

1. When you plug in the given points into the function, you generate two separate equations.
2. Use methods like substitution or elimination to solve these equations.

For our exercise, we used the elimination method. We subtracted the first equation from the second one to eliminate the variable 'b'. This helped us solve for 'm'. Once 'm' was found, we substituted its value back into one of the original equations to find 'b'. This guarantees that the solution satisfies both of the original equations, ensuring a comprehensive solution.
slope-intercept form
The slope-intercept form is a way of expressing linear functions. It's written as:

\(f(x) = mx + b\)

Here:
- \(m\) represents the slope of the line, which indicates how steep the line is.
- \(b\) is the y-intercept, where the line crosses the y-axis.

In our example, inserting given points into this form helps to create the equations we use to find 'm' and 'b'. The slope-intercept form is very useful in graphing because it immediately gives you the starting point of the line and how it progresses across the plane.

Slope, \(m\), is calculated by the difference in the y-values divided by the difference in the x-values between two points.
function notation
Function notation provides a way to describe a function mathematically. It is written as \(f(x)\), where:

- \(f\) indicates the function name.
- \(x\) represents the input of the function.

For a linear function like \(f(x) = mx + b\), you can insert specific values for 'x' to find corresponding values for 'y'.

In our exercise, \(f(3) = -3\) and \(f(-12) = -8\) use function notation to indicate specific pairs. This assists in creating equations to solve for unknowns 'm' and 'b'.

Understanding function notation is essential in math as it provides clarity and precision in describing relationships between variables. It helps in tracking input values and their corresponding output values efficiently, which is crucial for solving and graphing functions.

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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 farmer has 1200 acres of land and plans to plant corn and soybeans. The input cost (cost of seed, fertilizer, herbicide, and insecticide) for 1 acre for each crop is given in the table along with the cost of machinery and labor. The profit for 1 acre of each crop is given in the last column. $$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Input Cost } \\ \text { per Acre } \end{array} & \begin{array}{c} \text { Labor/Machinery } \\ \text { Cost per Acre } \end{array} & \begin{array}{c} \text { Profit } \\ \text { per Acre } \end{array} \\ \hline \text { Corn } & \$ 180 & \$ 80 & \$ 120 \\ \hline \text { Soybeans } & \$ 120 & \$ 100 & \$ 100 \\ \hline \end{array} $$ Suppose the farmer has budgeted a maximum of $$\$ 198,000$$ for input costs and a maximum of $$\$ 110,000$$ 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 were reversed between the two crops (that is, $$\$ 100$$ per acre for corn and $$\$ 120$$ per acre for soybeans), how many acres of each crop should be planted to maximize profit?

Use substitution to solve the system for the set of ordered triples \((x, y, \lambda)\) that satisfy the system. $$ \begin{array}{l} 2=2 \lambda x \\ 6=2 \lambda y \\ x^{2}+y^{2}=10 \end{array} $$

Solve the system. $$ \begin{array}{l} 3^{x}-9^{y}=18 \\ 3^{x}+3^{y}=30 \end{array} $$

Use the given constraints to find the maximum value of the objective function and the ordered pair \((x, y)\) that produces the maximum value. \(x \geq 0, y \geq 0\) \(3 x+4 y \leq 48\) \(2 x+y \leq 22\) \(y \leq 9\) a. Maximize: \(z=100 x+120 y\) b. Maximize: \(z=100 x+140 y\)
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