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Chapter 7: Problem 89

When solving a system of equations by substitution, how do you recognize that the system has no solution?

Short Answer

Expert verified
After substituting one equation into the other and simplifying, if a contradiction such as 0 = 5 is derived, it indicates the system of equations literally has no solution

Step by step solution

01

Choose an equation to solve

Firstly, select one equation to solve for any one of the variables.
02

Substitute

Substitute the solution from Step 1 into the other equation.
03

Simplify

Simplify the equation after substitution to see what you end up with.
04

Check the result

If you have arrived at a contradiction (for example, 0 = 5), it implies the system of equations has no solution.

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Most popular questions from this chapter

In Exercises 37-40, determine whether each ordered pair is a solution of the system of linear inequalities. \( \left\\{\begin{array}{l} x^2 + y^2 \ge 36\\\ -3x + y \le 10\\\ \dfrac{2}{3}x - y \ge 5\end{array}\right. \) (a) \( (-1, 7) \) (b) \( (-5, 1) \) (c) \( (6, 0) \) (d) \( (4, -8) \)

A merchant plans to sell two models of MP3 players at prices of \(\$ 225\) and \(\$ 250\) . The \(\$ 225\) model yields a profit of \(\$ 30\) per unit and the \(\$ 250\) model yields a profit of \(\$ 31\) per unit. The merchant estimates that the total monthly demand will not exceed 275 units. The merchant does not want to invest more than \(\$ 63,000\) in inventory for these products. What is the optimal inventory level for each model? What is the optimal profit?

In Exercises 55-60, use a graphing utility to graph the solution set of the system of inequalities. \( \left\\{\begin{array}{l} x^2 y \ge 1\\\ 0 < x \le 4\\\ \hspace{1cm} y \le 4\end{array}\right. \)

In Exercises 55-60, use a graphing utility to graph the solution set of the system of inequalities. \( \left\\{\begin{array}{l} y \ge x^4 - 2x^2 + 1\\\ y \le 1 - x^2\end{array}\right. \)

In Exercises 55-60, use a graphing utility to graph the solution set of the system of inequalities. \( \left\\{\begin{array}{l} y < -x^2 + 2x + 3\\\ y > x^2 - 4x + 3\end{array}\right. \)
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