Answered

Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

How many solutions will each system of linear equations have? Match the systems with the correct number of solutions.

1. [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex]
2. [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex]
3. [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex]

A. no solution
B. infinitely many solutions
C. one solution

Sagot :

GinnyAnswer
Alright, let's analyze each of the given systems of linear equations step-by-step and determine the number of solutions for each system.

### System 1:
[tex]\[ y = -2x + 5 \][/tex]
[tex]\[ 2x + y = -7 \][/tex]

Step-by-Step:
1. Substitute [tex]\( y = -2x + 5 \)[/tex] into [tex]\( 2x + y = -7 \)[/tex]:
[tex]\[ 2x + (-2x + 5) = -7 \][/tex]
2. Simplify the equation:
[tex]\[ 5 = -7 \][/tex]

This results in a contradiction. Therefore, the first system has no solution.

### System 2:
[tex]\[ y = x + 6 \][/tex]
[tex]\[ 3x - 3y = -18 \][/tex]

Step-by-Step:
1. Substitute [tex]\( y = x + 6 \)[/tex] into [tex]\( 3x - 3y = -18 \)[/tex]:
[tex]\[ 3x - 3(x + 6) = -18 \][/tex]
2. Simplify the equation:
[tex]\[ 3x - 3x - 18 = -18 \implies -18 = -18 \][/tex]

This is an identity, which means the equations are dependent, and every solution of the first equation is also a solution of the second equation. Thus, the second system has infinitely many solutions.

### System 3:
[tex]\[ y = -4x + 11 \][/tex]
[tex]\[ -6x + y = 11 \][/tex]

Step-by-Step:
1. Substitute [tex]\( y = -4x + 11 \)[/tex] into [tex]\( -6x + y = 11 \)[/tex]:
[tex]\[ -6x + (-4x + 11) = 11 \][/tex]
2. Simplify the equation:
[tex]\[ -6x - 4x + 11 = 11 \implies -10x + 11 = 11 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ -10x = 0 \implies x = 0 \][/tex]
4. Substitute [tex]\( x = 0 \)[/tex] back into [tex]\( y = -4x + 11 \)[/tex]:
[tex]\[ y = -4(0) + 11 = 11 \][/tex]

So, the solution is [tex]\( (0, 11) \)[/tex]. Therefore, the third system has one solution.

### Summary
- The system [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex] has no solution.
- The system [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex] has infinitely many solutions.
- The system [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex] has one solution.

Thus, the systems match with the number of solutions as follows:
- [tex]\( y = -2x + 5 \)[/tex] and [tex]\( 2x + y = -7 \)[/tex] → no solution
- [tex]\( y = x + 6 \)[/tex] and [tex]\( 3x - 3y = -18 \)[/tex] → infinitely many solutions
- [tex]\( y = -4x + 11 \)[/tex] and [tex]\( -6x + y = 11 \)[/tex] → one solution
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.