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
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.
what number should be on both sides of the equation -7x = 14 be divided by to solve for the value of x
a) Let's solve each equation step-by-step:
1. y = 5x - 4
This equation is already in slope-intercept form, y = mx + b, where m represents the slope and b represents the y-intercept.
The slope of this line is 5, indicating that for every 1 unit increase in x, the y-value increases by 5.
The y-intercept of this line is -4, meaning the line crosses the y-axis at the point (0, -4).
2. y = -2x + 3
Like the previous equation, this is also in slope-intercept form.
The slope of this line is -2, indicating that for every 1 unit increase in x, the y-value decreases by 2.
The y-intercept of this line is 3, meaning the line crosses the y-axis at the point (0, 3).
3. y = 3x + 1
Again, this equation is in slope-intercept form.
The slope of this line is 3, indicating that for every 1 unit increase in x, the y-value increases by 3.
The y-intercept of this line is 1, meaning the line crosses the y-axis at the point (0, 1).
4. 2y = 6x + 2
To rewrite this equation in slope-intercept form, we need to isolate y.
Divide both sides by 2: y = 3x + 1.
This equation is equivalent to equation 3. The only difference is coefficients on both sides.
5. y - 5x = 2
To put this equation in slope-intercept form, rearrange the terms: y = 5x + 2.
This equation has the same form as equation 1, with different coefficients.
6. y - 4x + 1 = 0
Rearranging the equation, we get y = 4x - 1.
This equation has the same form as equation 1 and equation 5, with different coefficients.
b. Let's graph all six equations:
Equation 1: y = 5x - 4
Equation 2: y = -2x + 3
Equation 3: y = 3x + 1
Equation 4: 2y = 6x + 2 (Equivalent to equation 3)
Equation 5: y = 5x + 2 (Equivalent to equation 1)
Equation 6: y = 4x - 1 (Equivalent to equations 1 and 5)
c. The two equations that form a system with infinite solutions are equations 4 and 3. They are equivalent equations, which means they represent the same line and will intersect at every point.
d. The two equations that form a system with no solution are equations 2 and 6. The lines represented by these equations are parallel and will never intersect.
e. The two equations that form a system with one solution are equations 1 and 5. These represent two distinct lines that intersect at a single point.
What can be noticed about equations in each category is that equations with infinite solutions are equivalent (same line), equations with no solution are parallel, and equations with one solution intersect at a single point.
To solve this problem, we will first need to rewrite the given equations in slope-intercept form (y = mx + b).
1. y = 5x – 4
This equation is already in slope-intercept form, with a slope of 5 and a y-intercept of -4.
2. y = –2x + 3
This equation is also in slope-intercept form, with a slope of -2 and a y-intercept of 3.
3. y = 3x + 1
Once again, this equation is in slope-intercept form, with a slope of 3 and a y-intercept of 1.
4. 2y = 6x + 2
To rewrite this equation in slope-intercept form, divide both sides of the equation by 2:
y = 3x + 1
This equation is equivalent to equation 3, so they represent the same line.
5. y – 5x = 2
To rewrite this equation in slope-intercept form, isolate y:
y = 5x + 2
This equation has a slope of 5 and a y-intercept of 2.
6. y - 4x + 1 = 0
To rewrite this equation in slope-intercept form, isolate y:
y = 4x - 1
This equation has a slope of 4 and a y-intercept of -1.
Now, let's graph these equations on a coordinate grid:
1. y = 5x – 4 (line with a slope of 5 and y-intercept of -4)
2. y = –2x + 3 (line with a slope of -2 and y-intercept of 3)
3. y = 3x + 1 (line with a slope of 3 and y-intercept of 1)
5. y = 5x + 2 (line with a slope of 5 and y-intercept of 2)
6. y = 4x - 1 (line with a slope of 4 and y-intercept of -1)
Now let's analyze the equations based on the given categories:
c. Circle the two equations that form a system with infinite solutions.
Equations that have the same slope and same y-intercept form a system with infinite solutions. In this case, we can circle equations 3 and 4 since they both have a slope of 3 and a y-intercept of 1.
d. Put a square around the two equations that form a system with no solution.
Equations that have different slopes and different y-intercepts form a system with no solution. In this case, we can put a square around equations 1 and 2 since they have different slopes and different y-intercepts.
e. Underline the two equations that form a system with one solution.
Equations that have different slopes and the same y-intercept form a system with one solution. In this case, we can underline equations 5 and 6 since they have different slopes (5 and 4) but the same y-intercept (2 and -1).
The equations that form a system with infinite solutions have the same slope and y-intercept. The equations that form a system with no solution have different slopes and different y-intercepts. The equations that form a system with one solution have different slopes and the same y-intercept.