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Chapter 12: Problem 35

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} 2 x+3 y=6 \\ x-y=\frac{1}{2} \end{array}\right. $$

Short Answer

Expert verified
(x, y) = (\frac{3}{2}, 1)

Step by step solution

01

Label the Equations

Label the given equations for easy reference. Equation (1): \(2x + 3y = 6\), Equation (2): \(x - y = \frac{1}{2}\).
02

Solve Equation (2) for x

Solve Equation (2) for \(x\): \(x = y + \frac{1}{2}\)
03

Substitute x into Equation (1)

Substitute \(x = y + \frac{1}{2}\) into Equation (1): \(2(y + \frac{1}{2}) + 3y = 6\)
04

Simplify the Equation

Distribute and combine like terms: \(2y + 1 + 3y = 6\). This simplifies to \(5y + 1 = 6\).
05

Solve for y

Subtract 1 from both sides to solve for \(y\): \(5y = 5\), then divide by 5: \(y = 1\)
06

Solve for x

Substitute \(y = 1\) back into Equation (2): \(x - 1 = \frac{1}{2}\), add 1 to both sides: \(x = \frac{3}{2}\)
07

Verify the Solution

Substitute \((x, y) = (\frac{3}{2}, 1)\) back into both original equations to verify the solution. \(2(\frac{3}{2}) + 3(1) = 6\): True, \( \frac{3}{2} - 1 = \frac{1}{2}\): True.
08

Conclusion

Since both equations are satisfied, the solution to the system is \((x, y) = (\frac{3}{2}, 1)\).

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Key Concepts

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

solving linear equations
Linear equations are fundamental in algebra and represent equations of the first degree. Each variable in a linear equation is raised to the power of one.
Common forms of linear equations include standard form, such as: \(ax + by = c\). Here, \a\, \b\, and \c\ are numerical coefficients. To solve a linear equation, isolate the variable on one side of the equation and simplify if needed. In the context of systems of equations, solving means finding values for the variables that satisfy all given equations simultaneously.
substitution method
The substitution method is one way to solve systems of linear equations. This involves substituting one equation into another to eliminate one variable, making it easier to solve.
Here's a step-by-step breakdown:
  • Step 1: Solve one of the equations for one variable in terms of the other variable.
  • Step 2: Substitute this expression into the other equation.
  • Step 3: Simplify and solve the resulting single-variable equation.
  • Step 4: Use the value obtained to find the other variable by substituting it back into any equation from the system.

This method systematically reduces the solution path, ensuring an organized approach to solving systems.
consistency of systems
A system of equations can be classified based on its solutions:
  • Consistent and independent: Exactly one solution.
  • Consistent and dependent: Infinitely many solutions.
  • Inconsistent: No solution.

To determine consistency, you can use either the substitution or elimination method to solve. If you reach a true statement, like \(0 = 0\), the system might be dependent. If you get a false statement, like \(0 = 5\), the system is inconsistent. In other cases, a unique solution means the system is consistent and independent.
solution verification
Verification is a crucial last step. After solving the system, substitute the found values back into the original equations to ensure they hold true.
For example, if you solve and get \x = \frac{3}{2}\ and \y = 1\, substituting into:
  • Equation (1): \(2(\frac{3}{2}) + 3(1) = 6\) simplifies to \6 = 6\.
  • Equation (2): \( \frac{3}{2} - 1 = \frac{1}{2}\) simplifies to \ \frac{1}{2} = \frac{1}{2}\.

Both validations confirm the correctness of the solution set. Verification helps avoid mistakes and provides confidence in the obtained solution.

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Most popular questions from this chapter

Three retired couples each require an additional annual income of \(\$ 2000\) per year. As their financial consultant, you recommend that they invest some money in Treasury bills that yield \(7 \%\), some money in corporate bonds that yield \(9 \%,\) and some money in "junk bonds" that yield \(11 \%\). Prepare a table for each couple showing the various ways that their goals can be achieved: (a) If the first couple has \(\$ 20,000\) to invest. (b) If the second couple has \(\$ 25,000\) to invest. (c) If the third couple has \(\$ 30,000\) to invest. (d) What advice would you give each couple regarding the amount to invest and the choices available?

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} x-y &=3 \\ -3 x+y &=1 \\ \end{array}\right.\\\ x=-2, y =-5 ;(-2,-5) \end{array} $$

Painting a House Three painters (Beth, Dan, and Edie), working together, can paint the exterior of a home in 10 hours (h). Dan and Edie together have painted a similar house in \(15 \mathrm{~h}\). One day, all three worked on this same kind of house for \(4 \mathrm{~h},\) after which Edie left. Beth and Dan required 8 more hours to finish. Assuming no gain or loss in efficiency, how long should it take each person to complete such a job alone?

Restaurant Management A restaurant manager wants to purchase 200 sets of dishes. One design costs \(\$ 25\) per set, and another costs \(\$ 45\) per set. If she has only \(\$ 7400\) to spend, how many sets of each design should she order?

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\left(\frac{18 x^{4} y^{5}}{27 x^{3} y^{9}}\right)^{3}\)
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