/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 a. Determine whether (5,-10) is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 1

a. Determine whether (5,-10) is a solution for the following system of equations: $$ \begin{array}{l} 4 x-3 y=50 \\ 2 x+2 y=5 \end{array} $$ b. Explain why (-10,5) is not a solution for the system in \(\operatorname{part}(a)\)

Short Answer

Expert verified
(5, -10) is not a solution because it does not satisfy the second equation. (-10, 5) is not a solution because it does not satisfy the first equation.

Step by step solution

01

Substitute (5, -10) into the first equation

To determine if the point (5, -10) is a solution to the system, substitute the values of x=5 and y=-10 into the first equation: 4x - 3y = 50 4(5) - 3(-10) = 50 20 + 30 = 50 50 = 50 (True)
02

Substitute (5, -10) into the second equation

Now, substitute x=5 and y=-10 into the second equation: 2x + 2y = 5 2(5) + 2(-10) = 5 10 - 20 = -10 -10 鈮� 5 (False)
03

Conclusion for part (a)

Since (5, -10) does not satisfy the second equation, (5, -10) is not a solution to the system of equations.
04

Check (-10, 5) for the first equation

Substitute x=-10 and y=5 into the first equation 4(-10) - 3(5) = 50 -40 - 15 = -55 -55 鈮� 50 (False)
05

Conclusion for part (b)

Since (-10, 5) does not satisfy the first equation, it is not a solution for the system given in part (a).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

solution verification
When working with systems of equations, it's important to verify whether a given point is a solution. This verification process involves substituting the x and y values of the point into each equation of the system.

For instance, the point (5, -10) needs to be checked by substituting into both equations: the first equation is checked as follows:
    \( 4(5) - 3(-10) = 50 \ \text{ which simplifies to:} \ 20 + 30 = 50 \)
and the second equation:
    \( 2(5) + 2(-10) = 5 \ \text{ simplifies to:} \ 10 - 20 = -10 \).
The result from the second equation clearly does not match up, showing that (5, -10) is not a solution.

Similarly, substituting (-10, 5) reveals that this point also fails the verification since
    \( 4(-10) - 3(5) = -55 \) <\ul>does not equal 50.

    Key Takeaway: Substituting points into each equation is crucial in confirming whether they are solutions to the system.
linear equations
Linear equations form the backbone of various algebraic concepts and are represented in the general form:
    \( ax + by = c \).
In simpler terms, this represents a line in a two-dimensional plane, where
    \( a \) and \( b \) are coefficients, \( x \) and \( y \) are variables, and \( c \) is a constant.
Each pair of linear equations, known as a system, can be solved to identify common intersection points.

For example, consider the equations from the original problem:
      \(4x - 3y = 50 \) and \( 2x + 2y = 5 \).
These represent two lines on a Cartesian plane, and finding solutions means identifying points where these lines intersect. Solution verification helps check if any specific points like (5,-10) or (-10,5) lie on these intersecting lines.
substitution method
The substitution method is a technique used to solve systems of linear equations. It involves solving one of the equations for one variable, and then substituting this expression into the other equation. This simplifies the system to a single equation with one variable, making it easier to solve.

Here鈥檚 how the substitution method could be applied to our example:
  • Step 1: Solve one of the equations for one variable. For instance, take the second equation: \( 2x + 2y = 5 \) and solve for \( y \):
      • \( y = \frac{5 - 2x}{2} \).
  • Step 2: Substitute this expression for \( y \) into the first equation:
      • \( 4x - 3\frac{5 - 2x}{2} = 50 \),
  • Step 3: Simplify and solve the resulting equation.


    • This method systematically breaks down the complexity of the system, highlighting intersections or solutions in a straightforward manner. It's crucial when direct verification isn't sufficient for understanding or solving problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Shade the region bounded by the inequalities $$ \begin{aligned} x+3 y & \leq 15 \\ 2 x+y & \leq 15 \\ x & \geq 0 \\ y & \geq 0 \end{aligned} $$

For the linear system \(\left\\{\begin{array}{l}x-y=5 \\ 2 x+y=1\end{array}\right.\) a. Graph the system. Estimate the solution for the system and then find the exact solution. b. Check that your solution satisfies both of the original equations.

The Food and Drug Administration labels suntan products with a sun protection factor (SPF) typically between 2 and 45 . Multiplying the SPF by the number of unprotected minutes you can stay in the sun without burning, you are supposed to get the increased number of safe sun minutes. For example, if you can stay unprotected in the sun for 30 minutes without burning and you apply a product with a SPF of \(10,\) then supposedly you can sun safely for \(30 \cdot 10=300\) minutes or 5 hours. Assume that you can stay unprotected in the sun for 20 minutes without burning. a. Write an equation that gives the maximum safe sun time \(T\) as a function of \(S,\) the sun protection factor (SPF). b. Graph your equation. What is the suggested domain for \(S ?\) c. Write an inequality that suggests times that would be unsafe to stay out in the sun. d. Shade in and label regions on the graph that indicate safe and unsafe regions. (Use two different shadings and remember to include boundaries.) e. How would the graph change if you could stay unprotected in the sun for 40 minutes? Note that you should be cautious about spending too much time in the sun. Factors such as water, wind, and sun intensity can diminish the effect of SPF products.

Calculate the solution(s), if any, to each of the following systems of equations. Use any method you like. $$ \begin{array}{ll} \text { a. } y=-1-2 x & \text { c. } y=2200 x-700 \\ y=13-2 x & y=1300 x+4700 \\ \text { b. } t=-3+4 w & \text { d. } 3 x=5 y \\ -12 w+3 t+9=0 & 4 y-3 x=-3 \end{array} $$

A Texas oil supplier sends out at most 10,000 barrels of oil per week. Two distributors need oil. Southern Oil needs at least 2000 barrels of oil per week and Regional Oil needs at least 5000 barrels of oil per week. a. Let \(S\) be the number of barrels of oil sent to Southern Oil and let \(R\) be the number of barrels sent to Regional Oil per week. Create a system of inequalities that describes all of the conditions. b. Graph the feasible region of the system. c. Choose a point inside the region that would satisfy the conditions and describe its meaning.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.