Math Problem Statement

3 x 13x1 minus− x 2x2 equals= negative 8−8 2 x 12x1 minus− 4 x 24x2 equals= negative 2−2 Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The unique solution is x 1 equalsx1=enter your response here and x 2 equalsx2=enter your response here. B. The system has infinitely many solutions. The solution is x 1 equalsx1=enter your response here and x 2x2equals=t. ​(Simplify your answer. Type an expression using t as the​ variable.) C. There is no solution.

Solution

We are given a system of two linear equations with two variables x1x_1 and x2x_2. Let's break down the system:

3x1−x2=−83x_1 - x_2 = -8 2x1−4x2=−22x_1 - 4x_2 = -2

Step 1: Solve the system using substitution or elimination.

Method 1: Elimination

We have the following two equations:

  1. 3x1−x2=−83x_1 - x_2 = -8
  2. 2x1−4x2=−22x_1 - 4x_2 = -2

Let's eliminate one of the variables by multiplying the first equation to match the second equation in terms of x1x_1.

Multiply the first equation by 2:

2(3x1−x2)=2(−8)2(3x_1 - x_2) = 2(-8) 6x1−2x2=−166x_1 - 2x_2 = -16

Now subtract the second equation from this:

(6x1−2x2)−(2x1−4x2)=−16−(−2)(6x_1 - 2x_2) - (2x_1 - 4x_2) = -16 - (-2) (6x1−2x2−2x1+4x2)=−16+2(6x_1 - 2x_2 - 2x_1 + 4x_2) = -16 + 2 4x1+2x2=−144x_1 + 2x_2 = -14

This simplifies to:

4x1+2x2=−144x_1 + 2x_2 = -14

This shows that there is an error in the solution method. I will

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Elimination Method

Formulas

Elimination Method: ax1 + bx2 = c
Linear Equation: ax + by = c

Theorems

Unique Solution Theorem: A system of two linear equations has a unique solution if the lines intersect at one point.
No Solution: Parallel lines never intersect, indicating no solution.

Suitable Grade Level

Grades 9-10

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