Question

which system of equations has a solution of approximately (-0.4, 1.1)?
○ $x + 3y = 3$ and $2x - y = -2$
○ $x - 3y = -3$ and $2x + y = 2$
○ $x - 3y = -3$ and $2x - y = -2$
○ $x + 3y = 3$ and $2x + y = 2$

Explanation:

Step1: Test the point (-0.4, 1.1) in the first equation of each option.

For option 1: \(x + 3y = -0.4 + 3\times1.1 = -0.4 + 3.3 = 2.9 eq3\); \(2x - y = 2\times(-0.4)-1.1 = -0.8 - 1.1 = -1.9 eq -2\).

Step2: Test for option 2: \(x - 3y = -0.4 - 3\times1.1 = -0.4 - 3.3 = -3.7

eq -3\); \(2x + y = 2\times(-0.4)+1.1 = -0.8 + 1.1 = 0.3 eq2\).

Step3: Test for option 3: \(x - 3y = -0.4 - 3\times1.1 = -0.4 - 3.3 = -3.7

eq -3\); \(2x - y = 2\times(-0.4)-1.1 = -0.8 - 1.1 = -1.9 eq -2\).

Step4: Test for option 4: \(x + 3y = -0.4 + 3\times1.1 = -0.4 + 3.3 = 2.9\approx3\); \(2x + y = 2\times(-0.4)+1.1 = -0.8 + 1.1 = 0.3\approx2\)? Wait, no, wait. Wait, let's recalculate option 3 and 4 again. Wait, option 3: \(x - 3y = -0.4 - 3\times1.1 = -3.7\), no. Option 4: \(x + 3y = -0.4 + 3.3 = 2.9\approx3\), \(2x + y = -0.8 + 1.1 = 0.3\)? No, wait, maybe I miscalculated. Wait, let's check option 3 again. Wait, option 3: \(x - 3y = -0.4 - 3(1.1) = -0.4 - 3.3 = -3.7\), not -3. Option 4: \(x + 3y = -0.4 + 3(1.1) = 2.9\approx3\), \(2x + y = 2(-0.4) + 1.1 = -0.8 + 1.1 = 0.3\)? No, that's not 2. Wait, maybe I made a mistake. Wait, let's check option 3 again. Wait, option 3: \(x - 3y = -3\), plug in (-0.4,1.1): -0.4 - 3(1.1) = -0.4 - 3.3 = -3.7, not -3. Option 4: \(x + 3y = 3\): -0.4 + 3(1.1) = 2.9 ≈ 3. \(2x + y = 2\): 2(-0.4) + 1.1 = -0.8 + 1.1 = 0.3 ≈ 2? No, 0.3 is not 2. Wait, maybe I messed up the options. Wait, let's check option 3 again. Wait, option 3: \(2x - y = -2\): 2(-0.4) - 1.1 = -0.8 - 1.1 = -1.9 ≈ -2. And \(x - 3y = -0.4 - 3(1.1) = -3.7\), no. Wait, option 4: \(x + 3y = 3\) (2.9≈3), \(2x + y = 2\) (0.3≈2? No). Wait, maybe the correct option is option 3? Wait, no, let's recalculate option 3: \(x - 3y = -0.4 - 3(1.1) = -3.7\), not -3. Option 4: \(x + 3y = -0.4 + 3.3 = 2.9≈3\), \(2x + y = -0.8 + 1.1 = 0.3≈2\)? No. Wait, maybe I made a mistake in the point. Wait, the point is (-0.4,1.1). Let's check option 3: \(x - 3y = -3\): -0.4 - 3(1.1) = -3.7, not -3. Option 4: \(x + 3y = 3\): 2.9≈3, \(2x + y = 2\): 0.3≈2? No. Wait, maybe the correct option is option 3? Wait, no, let's check option 3 again. Wait, \(2x - y = -2\): 2*(-0.4) - 1.1 = -0.8 - 1.1 = -1.9≈-2. \(x - 3y = -0.4 - 3.3 = -3.7\), not -3. Option 4: \(x + 3y = 2.9≈3\), \(2x + y = 0.3≈2\)? No. Wait, maybe the answer is option 3? Wait, no, let's check the options again. Wait, the correct option is option 3? Wait, no, let's do it properly.

Wait, let's take each option:

Option 1: \(x + 3y = 3\) and \(2x - y = -2\)

Test (-0.4,1.1) in \(x + 3y\): -0.4 + 3*1.1 = -0.4 + 3.3 = 2.9 ≈ 3 (close)

Test in \(2x - y\): 2*(-0.4) - 1.1 = -0.8 - 1.1 = -1.9 ≈ -2 (close)

Wait, 2.9≈3 and -1.9≈-2. So this is close. But earlier I thought it was not. Wait, maybe I miscalculated.

Option 1: \(x + 3y = 3\) (2.9≈3), \(2x - y = -2\) (-1.9≈-2). So this is the closest.

Wait, but earlier I thought option 1's first equation was 3, and 2.9 is close to 3, and -1.9 is close to -2. So option 1 is correct? Wait, no, let's check option 3: \(x - 3y = -0.4 - 3.3 = -3.7\), not -3. Option 4: \(2x + y = 0.3\), not 2. So option 1 is correct? Wait, no, the first option is \(x + 3y = 3\) and \(2x - y = -2\). Let's check:

\(x + 3y = -0.4 + 3(1.1) = 2.9 ≈ 3\)

\(2x - y = 2(-0.4) - 1.1 = -0.8 - 1.1 = -1.9 ≈ -2\)

Yes, this is the closest. So the correct option is the first one? Wait, no, the options are:

1. \(x + 3y = 3\) and \(2x - y = -2\)

2. \(x - 3y = -3\) and \(2x + y = 2\)

3. \(x - 3y = -3\) and \(2x - y = -2\)

4. \(x + 3y = 3\) and \(2x + y = 2\)

So testing (-0.4,1.1) in each:

Option 1:

Equation 1: -0.4 + 3(1.1) = 2.9 ≈ 3

Equation 2: 2(-0.4) - 1.1 = -1.9 ≈ -2

Option 2:

Equation 1: -0.4 - 3(1.1) = -3.7 ≈ -3? No, -3.7 is not close to -3.

Equation 2: 2(-0.4) + 1.1 = 0.3 ≈ 2? No.

Option 3:

Equation 1: -0.4 - 3(1.1) = -3.7 ≈ -3? No.

Equation 2: 2(-0.4) - 1.1 = -1.9 ≈ -2.

Option 4:

Equation 1: -0.4 + 3(1.1) = 2.9 ≈ 3

Equation 2: 2(-0.4) + 1.1 = 0.3 ≈ 2? No.

So the closest is option 1, where both equations are approximately satisfied.

Answer:

A. \(x + 3y = 3\) and \(2x - y = -2\)