Answer :
To determine which of the given points lie on the line defined by the slope and y-intercept, we first need to form the equation of the line. The slope-intercept form of a line is given by the equation:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Plugging in the given values:
[tex]\[ y = -2x - 3 \][/tex]
Next, we will check each of the given points to see if they satisfy this equation.
1. Point [tex]\( (-2, 0) \)[/tex]:
[tex]\[ y = -2(-2) - 3 = 4 - 3 = 1 \][/tex]
This results in [tex]\( y = 1 \)[/tex], not [tex]\( y = 0 \)[/tex], so [tex]\( (-2, 0) \)[/tex] does not lie on the line.
2. Point [tex]\( (-1.5, 0) \)[/tex]:
[tex]\[ y = -2(-1.5) - 3 = 3 - 3 = 0 \][/tex]
This results in [tex]\( y = 0 \)[/tex], so [tex]\( (-1.5, 0) \)[/tex] lies on the line.
3. Point [tex]\( (-6, 9) \)[/tex]:
[tex]\[ y = -2(-6) - 3 = 12 - 3 = 9 \][/tex]
This results in [tex]\( y = 9 \)[/tex], so [tex]\( (-6, 9) \)[/tex] lies on the line.
4. Point [tex]\( (2, -3) \)[/tex]:
[tex]\[ y = -2(2) - 3 = -4 - 3 = -7 \][/tex]
This results in [tex]\( y = -7 \)[/tex], not [tex]\( y = -3 \)[/tex], so [tex]\( (2, -3) \)[/tex] does not lie on the line.
5. Point [tex]\( (3, -9) \)[/tex]:
[tex]\[ y = -2(3) - 3 = -6 - 3 = -9 \][/tex]
This results in [tex]\( y = -9 \)[/tex], so [tex]\( (3, -9) \)[/tex] lies on the line.
6. Point [tex]\( (-4, 4) \)[/tex]:
[tex]\[ y = -2(-4) - 3 = 8 - 3 = 5 \][/tex]
This results in [tex]\( y = 5 \)[/tex], not [tex]\( y = 4 \)[/tex], so [tex]\( (-4, 4) \)[/tex] does not lie on the line.
Therefore, the points that lie on the line [tex]\( y = -2x - 3 \)[/tex] are:
[tex]\[ (-1.5, 0), (-6, 9), \text{ and } (3, -9) \][/tex]
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Plugging in the given values:
[tex]\[ y = -2x - 3 \][/tex]
Next, we will check each of the given points to see if they satisfy this equation.
1. Point [tex]\( (-2, 0) \)[/tex]:
[tex]\[ y = -2(-2) - 3 = 4 - 3 = 1 \][/tex]
This results in [tex]\( y = 1 \)[/tex], not [tex]\( y = 0 \)[/tex], so [tex]\( (-2, 0) \)[/tex] does not lie on the line.
2. Point [tex]\( (-1.5, 0) \)[/tex]:
[tex]\[ y = -2(-1.5) - 3 = 3 - 3 = 0 \][/tex]
This results in [tex]\( y = 0 \)[/tex], so [tex]\( (-1.5, 0) \)[/tex] lies on the line.
3. Point [tex]\( (-6, 9) \)[/tex]:
[tex]\[ y = -2(-6) - 3 = 12 - 3 = 9 \][/tex]
This results in [tex]\( y = 9 \)[/tex], so [tex]\( (-6, 9) \)[/tex] lies on the line.
4. Point [tex]\( (2, -3) \)[/tex]:
[tex]\[ y = -2(2) - 3 = -4 - 3 = -7 \][/tex]
This results in [tex]\( y = -7 \)[/tex], not [tex]\( y = -3 \)[/tex], so [tex]\( (2, -3) \)[/tex] does not lie on the line.
5. Point [tex]\( (3, -9) \)[/tex]:
[tex]\[ y = -2(3) - 3 = -6 - 3 = -9 \][/tex]
This results in [tex]\( y = -9 \)[/tex], so [tex]\( (3, -9) \)[/tex] lies on the line.
6. Point [tex]\( (-4, 4) \)[/tex]:
[tex]\[ y = -2(-4) - 3 = 8 - 3 = 5 \][/tex]
This results in [tex]\( y = 5 \)[/tex], not [tex]\( y = 4 \)[/tex], so [tex]\( (-4, 4) \)[/tex] does not lie on the line.
Therefore, the points that lie on the line [tex]\( y = -2x - 3 \)[/tex] are:
[tex]\[ (-1.5, 0), (-6, 9), \text{ and } (3, -9) \][/tex]
