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Without graphing, determine if the two lines are intersecting, parallel or the same line. (If the equation is not in slope-Intercept form, show all work to convert equation to slope-intercept form)
y=4x-5
(a) 8x + y = 3
Intersecting
B parallel
same lines
(b) The system of equation will have one
solution(s).

Verified Answer :

GinnyAnswer

To determine the relation between the two lines, we first need to see if both equations are in slope-intercept form, which is \(\( y = mx + b \)\), where \(\( m \)\) represents the slope and \(\( b \)\) represents the y-intercept.

The first equation is already in the slope-intercept form:
\(\[ y = 4x - 5 \]\)

This equation has a slope of \(\( m = 4 \)\) and a y-intercept of \(\( b = -5 \)\).

The second equation is:
\(\[ 8x + y = 3 \]\)

To convert it into slope-intercept form, we isolate \(\( y \)\):
\(\[ y = -8x + 3 \]\)

Now it's in the slope-intercept form with a slope \(\( m = -8 \)\) and y-intercept \(\( b = 3 \)\).

The slopes of the two lines are:
- Slope of the first line: \(\( m = 4 \)\)
- Slope of the second line: \(\( m = -8 \)\)

These slopes are not equal to each other. Since the slopes are different, the lines are neither parallel nor the same line; they are intersecting. This means:
- The two lines are intersecting, which answers part (a).

Now let's consider part (b) regarding the number of solutions:
Because the lines are intersecting, they meet at exactly one point. Therefore, the system of equations will have exactly one solution. Hence for part (b), the answer is:
- The system of equations will have one solution.

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