Question

select the correct answer from the drop-down menu. consider the equations y = |x - 1| and y = 3x + 2. the approximate solution of this system of equations is .

Explanation:

Step1: Analyze the absolute value function

The equation \( y = |x - 1| \) can be split into two cases: when \( x - 1\geq0 \) (i.e., \( x\geq1 \)), \( y = x - 1 \); when \( x - 1<0 \) (i.e., \( x<1 \)), \( y = -x + 1 \).

Step2: Solve for \( x\geq1 \) case

Set \( x - 1 = 3x + 2 \) (since for \( x\geq1 \), \( y = x - 1 \) and \( y = 3x + 2 \)).
Subtract \( x \) from both sides: \( -1 = 2x + 2 \).
Subtract 2 from both sides: \( -3 = 2x \).
Then \( x = -\frac{3}{2}=-1.5 \). But this value is less than 1, so it doesn't satisfy the \( x\geq1 \) condition, so we discard this solution.

Step3: Solve for \( x<1 \) case

Set \( -x + 1 = 3x + 2 \) (since for \( x<1 \), \( y = -x + 1 \) and \( y = 3x + 2 \)).
Add \( x \) to both sides: \( 1 = 4x + 2 \).
Subtract 2 from both sides: \( -1 = 4x \).
Then \( x = -\frac{1}{4}=-0.25 \).

Step4: Find the corresponding \( y \)-value

Substitute \( x = -0.25 \) into \( y = 3x + 2 \): \( y = 3(-0.25)+2=-0.75 + 2 = 1.25 \).

So the solution to the system is \( x\approx - 0.25 \), \( y\approx1.25 \) (or in ordered pair form \((-0.25, 1.25)\)).

Answer:

The approximate solution of the system of equations \( y = |x - 1| \) and \( y = 3x + 2 \) is \((-0.25, 1.25)\) (or in the form of the ordered pair with \( x\approx - 0.25 \) and \( y\approx1.25 \)).