To graph all six equations, we will choose x-values and solve for y. Then, we will plot the points on the coordinate grid.
1. y = 5x – 4
For x = 0, y = -4, giving us the point (0,-4).
For x = 1, y = 5(1) - 4 = 1, giving us the point (1,1).
Joining these points, we get the line:
2. y = -2x + 3
For x = 0, y = 3, giving us the point (0,3).
For x = 1, y = -2(1) + 3 = 1, giving us the point (1,1).
Joining these points, we get the line:
3. y = 3x + 1
For x = 0, y = 1, giving us the point (0,1).
For x = 1, y = 3(1) + 1 = 4, giving us the point (1,4).
Joining these points, we get the line:
4. 2y = 6x + 2
We can rearrange the equation to y = 3x + 1.
This equation is the same as equation 3 and represents the same line.
5. y - 5x = 2
Rearranging the equation, we get y = 5x + 2.
For x = 0, y = 2, giving us the point (0,2).
For x = 1, y = 5(1) + 2 = 7, giving us the point (1,7).
Joining these points, we get the line:
6. y - 4x + 1 = 0
Rearranging the equation, we get y = 4x - 1.
For x = 0, y = -1, giving us the point (0,-1).
For x = 1, y = 4(1) - 1 = 3, giving us the point (1,3).
Joining these points, we get the line:
c. The two equations that form a system with infinite solutions are:
- 3. y = 3x + 1
- 4. 2y = 6x + 2 (or y = 3x + 1, as they represent the same line)
d. The two equations that form a system with no solution are:
- 1. y = 5x – 4
- 2. y = -2x + 3
e. The two equations that form a system with one solution are:
- 5. y - 5x = 2
- 6. y - 4x + 1 = 0
Observation:
- Equations that form a system with infinite solutions are essentially the same line or have the same slope-intercept form.
- Equations that form a system with no solution have different slopes and do not intersect.
- Equations that form a system with one solution have different slopes and intersect at a single point.
1. y=5x–4
2. y=–2x+3
3. y=3x+1
4. 2y=6x+2
5. y–5x=+2
6. y–4x+1=0
b. Graph all six equations on the coordinate grid and label them.
c. Circle the two equations that form a system with infinite solutions.
d. Put a square around the two equations that form a system with no solution.
e. Underline the two equations that form a system with one solution.
What do you notice about equations that are in each of these categories? Use
complete sentences to answer the questions that follow
2 answers
what number should be on both sides of the equation -7x = 14 be divided by to solve for the value of x