Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 51

Chapter 6, Problem 51

Solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0 and b ≠ 0. For the linear function f(x) = mx + b, f(−2) = 11 and ƒ(3) = -9. Find m and b.

Verified step by step guidance

Start with the general form of the linear function: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Use the given values to create two equations by substituting the points into the function. For $f(-2) = 11$, substitute $x = -2$ and $f(x) = 11$ to get: $11 = m(-2) + b$.

Similarly, for $f(3) = -9$, substitute $x = 3$ and $f(x) = -9$ to get: $-9 = m(3) + b$.

Now you have a system of two linear equations: $11 = -2m + b$ $-9 = 3m + b$

Solve this system of equations simultaneously to find the values of $m$ and $b$. You can use either substitution or elimination method.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Linear Functions

A linear function is an algebraic expression of the form f(x) = mx + b, where m represents the slope and b the y-intercept. It graphs as a straight line, and understanding this form helps in identifying how changes in x affect f(x).

Slope of a Line

The slope (m) measures the rate of change of the function, calculated as the change in y divided by the change in x between two points. It indicates the steepness and direction of the line, essential for determining m from given points.

Solving Systems of Equations

Given two points on the line, substituting them into f(x) = mx + b creates two equations with two unknowns (m and b). Solving this system simultaneously allows finding the exact values of m and b.

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Related Practice

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

$\begin{cases}\)(x - 1)^2 + (y + 1)^2 < 25 \$x - 1)^2 + (y + 1)^2 \(\geq$ 16\(\end{cases}$

Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x²+y²<16, y≥2^x

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Textbook Question

In Exercises 47–52, solve each system by the method of your choice. $\begin{cases}\]\frac{3}{x^2}\) + $\frac{1}{y^2}$ = 7 $\frac{5}{x^2}$ - $\frac{2}{y^2}$ = -3\(\end{cases}$

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Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x²+y²≤1, y−x²>0

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Textbook Question

Find the value of the objective function z = 2x + 3y at each corner of the graphed region shown. What is the maximum value of the objective function? What is the minimum value of the objective function?

Textbook Question

Find the partial fraction decomposition for 1/x(x+1) and use the result to find the following sum:

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