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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 2 (page 793)

In Problems 1鈥�10, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 3 x - 4 y = 4 \\ x - 3 y = \frac{1}{2} \end{cases}$

Short Answer

Expert verified

The solution of the system is $(2, \frac{1}{2})$ .

Step by step solution

01

Step 1. Given Information

We are given a system of equations $\{\begin{cases} 3 x - 4 y = 4 \\ x - 3 y = \frac{1}{2} \end{cases}$ .

We need to solve the system using the method of substitution or the method of elimination.

02

Step 2. Substitute the value of x in the first equation

Solving the second equation $x - 3 y = \frac{1}{2}$ for $x$ we get

$\begin{array}{rcl} x - 3 y + 3 y & = & \frac{1}{2} + 3 y \\ x & = & 3 y + \frac{1}{2} \end{array}$

Now substitute $3 y + \frac{1}{2}$ for $x$ in the first equation $3 x - 4 y = 4$ and solve for $y$

$\begin{array}{rcl} 3 (3 y + \frac{1}{2}) - 4 y & = & 4 \\ 9 y + \frac{3}{2} - 4 y & = & 4 \\ 5 y & = & 4 - \frac{3}{2} \\ 5 y & = & \frac{5}{2} \\ y & = & \frac{1}{2} \end{array}$

03

Step 3. Solve for the value of x

In the equation $x = 3 y + \frac{1}{2}$ substitute $\frac{1}{2}$ for role="math" $y$

$x = 3 路 \frac{1}{2} + \frac{1}{2} x = \frac{3}{2} + \frac{1}{2} x = \frac{4}{2} x = 2$

So the solution of the system is $(2, \frac{1}{2})$ .

04

Step 4. Check the solution

Substitute $2$ for $x$ and $\frac{1}{2}$ for $y$ in both the equations.

$\begin{array}{rcl} 3 x - 4 y & = & 4 \\ 3 路 2 - 4 路 \frac{1}{2} & = & 4 \\ 6 - 2 & = & 4 \\ 4 & = & 4 \\ x - 3 y & = & \frac{1}{2} \\ 2 - 3 路 \frac{1}{2} & = & \frac{1}{2} \\ \frac{4 - 3}{2} & = & \frac{1}{2} \\ \frac{1}{2} & = & \frac{1}{2} \end{array}$

So both the equations are satisfied. Thus the found solution is correct.

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Most popular questions from this chapter

Constructing a Box: A rectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2-centimeter square from each corner and turning up the sides. See the figure. If the box is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Solve each system of equations. If the system has no solution, say that it is inconsistent

x + 2y - z = -3

2x - 4y + z = -7

-2x + 2y - 3z = 4

Solve each system of equations. If the system has no solution, say that it is inconsistent.
$\{\begin{cases} 3 x - 6 y = 2 \\ 5 x + 4 y = 1 \end{cases}$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = \sqrt{4 - x^{2}}; y = 2 x + 4$

In Problems 5鈥�24, graph each equation of the system. Then solve the system to find the points of intersection.

$y = \sqrt{x}; y = 6 - x$

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