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Consider the equations [tex][tex]$y=|x-1|$[/tex][/tex] and [tex][tex]$y=3x+2$[/tex][/tex].

The approximate solution of this system of equations is [tex][tex]$\square$[/tex][/tex]



Answer :

To find the solution to the system of equations given by [tex]\( y = |x - 1| \)[/tex] and [tex]\( y = 3x + 2 \)[/tex], we need to analyze each equation and see where they intersect.

### Step 1: Analyze the Equations
Equation 1: [tex]\( y = |x - 1| \)[/tex]
This represents a V-shaped graph with the vertex at [tex]\( (1, 0) \)[/tex]. It breaks into two linear parts:

1. For [tex]\( x \ge 1 \)[/tex]: [tex]\( y = x - 1 \)[/tex]
2. For [tex]\( x < 1 \)[/tex]: [tex]\( y = -(x - 1) = -x + 1 \)[/tex]

Equation 2: [tex]\( y = 3x + 2 \)[/tex]
This represents a straight line with a slope of 3 and a y-intercept of 2.

### Step 2: Solve for Intersections

Case 1: [tex]\( x \ge 1 \)[/tex]

For [tex]\( x \ge 1 \)[/tex], [tex]\( y = x - 1 \)[/tex]:
Set [tex]\( x - 1 = 3x + 2 \)[/tex]:

[tex]\[ x - 1 = 3x + 2 \][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[ x - 3x = 2 + 1 \\ -2x = 3 \\ x = -\frac{3}{2} \][/tex]

Since [tex]\( x \ge 1 \)[/tex], [tex]\( x = -\frac{3}{2} \)[/tex] is not valid in this interval.

Case 2: [tex]\( x < 1 \)[/tex]

For [tex]\( x < 1 \)[/tex], [tex]\( y = -x + 1 \)[/tex]:
Set [tex]\( -x + 1 = 3x + 2 \)[/tex]:

[tex]\[ -x + 1 = 3x + 2 \][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[ 1 - 2 = 3x + x \\ -1 = 4x \\ x = -\frac{1}{4} \][/tex]

Since [tex]\( x < 1 \)[/tex], [tex]\( x = -\frac{1}{4} \)[/tex] is valid here.

### Step 3: Find the corresponding [tex]\( y \)[/tex]-value

Substituting [tex]\( x = -\frac{1}{4} \)[/tex] into [tex]\( y = 3x + 2 \)[/tex]:

[tex]\[ y = 3\left(-\frac{1}{4}\right) + 2 \\ y = -\frac{3}{4} + 2 \\ y = \frac{5}{4} \][/tex]

### Conclusion

The approximate solution to the system of equations [tex]\( y = |x - 1| \)[/tex] and [tex]\( y = 3x + 2 \)[/tex] is: [tex]\( \left( -\frac{1}{4}, \frac{5}{4} \right) \)[/tex].

So, you should select:

[tex]\[ \boxed{\left( -\frac{1}{4}, \frac{5}{4} \right)} \][/tex]