Part A: Explain why the x-coordinates of the points where the graphs of the equations y = 4x and y = 2x−2 intersect are the solutions of the equation 4x = 2x−2.
Part B: Make tables to find the solution to 4x = 2x−2. Take the integer values of x between −3 and 3.
Part C: How can you solve the equation 4x = 2x−2 graphically?
Part A:
To explain why the x-coordinates of the points where the graphs of the equations y = 4x and y = 2x - 2 intersect are the solutions of the equation 4x = 2x - 2, we need to consider the concept of intersection between two graphs.
When two graphs intersect, it means that they have a common point. In the case of these two equations, we are looking for values of x for which both equations have the same y-value. By setting the y-values of both equations equal to each other, we can find the x-values that satisfy this condition.
So, let's set the equations equal to each other and solve for x:
4x = 2x - 2
Subtracting 2x from both sides, we get:
2x = -2
Now, we divide both sides of the equation by 2:
x = -1
The x-coordinate of the intersection point is -1. Therefore, the solution to the equation 4x = 2x - 2 is x = -1.
Part B:
To find the solution to the equation 4x = 2x - 2, we can create a table and evaluate the equation for different integer values of x between -3 and 3.
| x | 4x | 2x - 2 |
|------|------|---------|
| -3 | -12 | -8 |
| -2 | -8 | -6 |
| -1 | -4 | -4 |
| 0 | 0 | -2 |
| 1 | 4 | 0 |
| 2 | 8 | 2 |
| 3 | 12 | 4 |
By observing the table, we can see that the value of x that satisfies the equation 4x = 2x - 2 is x = -1.
Part C:
To solve the equation 4x = 2x - 2 graphically, you can plot the graphs of both equations on the same coordinate system and find the point of intersection.
1. Start by plotting the graph of y = 4x, which is a straight line passing through the origin (0,0) with a slope of 4. You can plot a few points on the line to help you draw it accurately.
2. Next, plot the graph of y = 2x - 2, which is another straight line with a y-intercept of -2 and a slope of 2. Again, plot a few points to help you draw the line.
3. Look for the point where the two lines intersect. This point represents the solution to the equation 4x = 2x - 2. The x-coordinate of this point is the solution to the equation.
Alternatively, you can use a graphing calculator or an online graphing tool to plot the graphs of the two equations and find the point of intersection.
Part A:
To find the points where the graphs of the equations y = 4x and y = 2x−2 intersect, we need to find the x-coordinates of those points. This means we need to set the two equations equal to each other and solve for x.
So, the equation is set up as 4x = 2x−2.
Part B:
To find the solution to 4x = 2x−2, we can make a table by substituting different integer values of x between -3 and 3 into the equation and finding the corresponding values of 4x and 2x−2.
Let's make the following table:
| x | 4x | 2x−2 |
| -3 | -12 | -8 |
| -2 | -8 | -6 |
| -1 | -4 | -4 |
| 0 | 0 | -2 |
| 1 | 4 | 0 |
| 2 | 8 | 2 |
| 3 | 12 | 4 |
From the table, we can see that when x = 1, the values of 4x and 2x−2 are equal, which satisfies the equation 4x = 2x−2. Therefore, x = 1 is the solution to the equation.
Part C:
To solve the equation 4x = 2x−2 graphically, we can plot the graphs of the two equations y = 4x and y = 2x−2 on a coordinate plane and identify the point where they intersect.
Start by plotting the points for each equation and then connect them to form the lines. The intersection point represents the solution to the equation.
The graph of y = 4x is a straight line passing through the origin (0, 0) with a slope of 4.
The graph of y = 2x−2 is a straight line passing through the y-intercept of (0, -2) with a slope of 2.
Now, plot these points and draw the lines on the coordinate plane. The point where the two lines intersect is the solution to the equation.
Note: If the lines are not visually intersecting, you can use a ruler to get a more accurate estimation of the intersection point.
Once you have identified the intersection point, you can read off the x-coordinate to find the solution to the equation.