Question

Which statement describes the graph of the system of equations below?

1.5x + 0.2y = 2.68

1.6x + 0.3y = 2.98

The lines are parallel.
The lines overlap at all points.
The lines intersect at (1.6,1.4).
The lines intersect at (3.1,0.5).

Answer 1

To determine the relationship between the two lines represented by the given equations, we can analyze their slopes and intercepts.

1. **Find the slope of each equation in the slope-intercept form (y = mx + b):**

**For the first equation:**
\[ 1.5x + 0.2y = 2.68 \]

Rearranging this gives:
\[ 0.2y = -1.5x + 2.68 \]
\[ y = -7.5x + 13.4 \]

The slope (m) of the first line is -7.5.

**For the second equation:**
\[ 1.6x + 0.3y = 2.98 \]

Rearranging this gives:
\[ 0.3y = -1.6x + 2.98 \]
\[ y = -\frac{16}{3}x + \frac{2.98}{0.3} \]
Calculating the constant:
\[ \frac{2.98}{0.3} \approx 9.9333 \]
So,
\[ y \approx -5.3333x + 9.9333 \]

The slope (m) of the second line is approximately -5.33.

2. **Compare the slopes:**
The slopes are different (-7.5 and -5.33), so the lines are not parallel.

3. **Check if there is a point of intersection:**
To find if they intersect, we can try solving the two equations together. We can set them equal to find the \(x\) and \(y\) values.

Solving the system of equations:
1. \( 1.5x + 0.2y = 2.68 \) (Multiply by 10) → \( 15x + 2y = 26.8 \)
2. \( 1.6x + 0.3y = 2.98 \) (Multiply by 10) → \( 16x + 3y = 29.8 \)

Now we can use the method of elimination or substitution, but instead, we'll use a quick numerical check:

1. If we check the proposed intersection points:
- For (1.6, 1.4):
- First equation: \( 1.5(1.6) + 0.2(1.4) = 2.4 + 0.28 = 2.68 \) (True)
- Second equation: \( 1.6(1.6) + 0.3(1.4) = 2.56 + 0.42 = 2.98 \) (True)
- Therefore, they indeed intersect at (1.6, 1.4).

2. Checking the second proposed point (3.1, 0.5):
- First equation: \( 1.5(3.1) + 0.2(0.5) = 4.65 + 0.1 = 4.75\) (False)
- This point does not satisfy the first equation.

So the valid conclusion is:

**The lines intersect at (1.6, 1.4).**