To find the best estimate to the solution of the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
y = \frac{1}{4}x - 2 \\
y = -2x + 3
\end{array}
\right.
\][/tex]
we will solve these equations step-by-step.
### Step 1: Find the Intersection Point
To solve the system of equations, we need to find the point where both lines intersect. This means we have to set the equations equal to each other:
[tex]\[
\frac{1}{4}x - 2 = -2x + 3
\][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
Combine the terms involving [tex]\( x \)[/tex] on one side and the constants on the other side:
[tex]\[
\frac{1}{4}x + 2x = 3 + 2
\][/tex]
[tex]\[
\frac{1}{4}x + \frac{8}{4}x = 5
\][/tex]
[tex]\[
\frac{9}{4}x = 5
\][/tex]
Multiply both sides by 4 to clear the fraction:
[tex]\[
9x = 20
\][/tex]
Divide both sides by 9:
[tex]\[
x = \frac{20}{9} \approx 2.22
\][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Substitute [tex]\( x = \frac{20}{9} \)[/tex] back into either of the original equations. Let's use [tex]\( y = \frac{1}{4}x - 2 \)[/tex]:
[tex]\[
y = \frac{1}{4} \left(\frac{20}{9}\right) - 2
\][/tex]
[tex]\[
y = \frac{20}{36} - 2 = \frac{5}{9} - 2
\][/tex]
Convert 2 to a fraction with denominator 9:
[tex]\[
y = \frac{5}{9} - \frac{18}{9} = \frac{5 - 18}{9} = -\frac{13}{9} \approx -1.44
\][/tex]
### Step 4: Compare with Given Options
We found the point of intersection to be approximately [tex]\( \left(2.22, -1.44\right) \)[/tex].
Now, let’s compare this with the given options:
- [tex]\( (0, -2) \)[/tex]
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (1.5, 0) \)[/tex]
- [tex]\( (2.2, -1.4) \)[/tex]
The option [tex]\( (2.2, -1.4) \)[/tex] is the closest to [tex]\( (2.22, -1.44) \)[/tex].
### Conclusion
The best estimate to the solution of the given system of equations is [tex]\( (2.2, -1.4) \)[/tex].
