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 29

Chapter 6, Problem 29

In Exercises 19–30, solve each system by the addition method. 3x = 4y + 1 3y = 1 - 4x

Verified step by step guidance

Rewrite both equations in standard form (Ax + By = C) to prepare for the addition method. For the first equation, subtract 4y and 1 from both sides to get

3x - 4 y = 1

. For the second equation, add 4x to both sides and subtract 1 from both sides to get

4 x + 3 y = 1

Align the two equations for addition:

3x - 4y = 1

and

4x + 3y = 1

Multiply each equation by a suitable number so that the coefficients of either

x

y

are opposites. For example, multiply the first equation by 3 and the second equation by 4 to align the coefficients of

y

3(3x - 4y) = 3(1)

and

4(4x + 3y) = 4(1)

Add the two resulting equations to eliminate

y

. This will give you an equation with only

x

. Solve this equation for

x

Substitute the value of

x

back into one of the original equations to solve for

y

. This completes the solution to the system.

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Key Concepts

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

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution is the set of variable values that satisfy all equations simultaneously. Understanding how to interpret and manipulate these equations is essential for solving the system.

Addition Method (Elimination Method)

The addition method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the remaining variable. This method requires aligning terms and possibly multiplying equations to create opposite coefficients for one variable.

Rearranging Equations

Before applying the addition method, equations often need to be rearranged into standard form (Ax + By = C). This makes it easier to identify coefficients and perform elimination. Rearranging involves moving all terms to one side and simplifying.

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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}\)2x - 5y $\leq$ 10 \\3x - 2y > 6\(\end{cases}$

Textbook Question

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

$\begin{cases}\)3x + 6y $\leq$ 6 \\2x + y $\leq$ 8\(\end{cases}$

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Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)2x^2 + y^2 = 18 $\xy$ = 4\(\end{cases}$

Textbook Question

In Exercises 29–42, solve each system by the method of your choice. $\begin{cases}\)3x^2 + 4y^2 = 16 \\2x^2 - 3y^2 = 5\(\end{cases}$

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Textbook Question

Write the partial fraction decomposition of each rational expression. 5x² -6x+7/(x − 1) (x² + 1)

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Textbook Question

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

$\begin{cases}\)y > 2x - 3 $\y$ < -x + 6\(\end{cases}$