Question

question 2 of 14 step 1 of 1 solve the following system of linear equations by addition. indicate whether the given system of linear equations is consistent, inconsistent, or dependent. if the system is consistent, find the solution. (\begin{cases}-4x + 2y=14\\3x + y = 2end{cases}) answer 5 points selecting an option will enable input for any required text boxes. if the selected option does not have any associated text boxes, then no further input is required. consistent ( ) inconsistent dependent

Explanation:

Step1: Multiply first - equation to eliminate y

Multiply the equation \(3x + y=2\) by \(2\) to get \(6x + 2y = 4\).

Step2: Subtract equations

Subtract the new - formed equation \(6x + 2y = 4\) from \(-4x+2y = 14\). \((-4x + 2y)-(6x + 2y)=14 - 4\). Simplify the left - hand side: \(-4x+2y - 6x - 2y=-10x\), and the right - hand side is \(10\). So, \(-10x = 10\).

Step3: Solve for x

Divide both sides of \(-10x = 10\) by \(-10\), we get \(x=-1\).

Step4: Substitute x into the first equation to solve for y

Substitute \(x = - 1\) into \(3x + y=2\), then \(3\times(-1)+y = 2\), which simplifies to \(-3 + y=2\). Add \(3\) to both sides, we get \(y = 5\).

Step5: Check consistency

Since we found a unique solution \((x=-1,y = 5)\), the system is consistent.

Answer:

Consistent; \(x=-1,y = 5\)