Question

Without​ graphing, answer the questions for the system of equations below.

2 x plus y2x+y
​=
negative 3−3
negative x plus 5 y−x+5y
​=
1010
Question content area bottom
Part 1
a.
Are the graphs of the given equations identical​ lines, parallel​ lines, or lines intersecting at a single​ point?
Identical lines
Parallel lines
Intersecting lines

Answer 1

To determine the relationship between the two equations, we can analyze their slopes and intercepts. Let's first rewrite the given equations in slope-intercept form (y = mx + b).

1. The first equation is:
\[
2x + y = -3
\]
Rearranging gives us:
\[
y = -2x - 3
\]
Here, the slope \( m_1 = -2 \).

2. The second equation is:
\[
-x + 5y = 10
\]
Rearranging gives us:
\[
5y = x + 10
\]
Dividing everything by 5 gives us:
\[
y = \frac{1}{5}x + 2
\]
Here, the slope \( m_2 = \frac{1}{5} \).

Now we can compare the slopes:
- The first equation has a slope of \(-2\).
- The second equation has a slope of \(\frac{1}{5}\).

Since the slopes are different, the lines are **not parallel** and will intersect at a single point.

Hence, to answer Part 1:
- The correct option is **Intersecting lines**.