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?
the graphs intersect where both x and y are the same on both graphs. That is, given any particular value of x, you want
4x = 2x-2
the table part is easy. To solve graphically, draw the graphs and note where they intersect.
Part A: To find the x-coordinates where the graphs of the equations y = 4x and y = 2x - 2 intersect, we need to set the two equations equal to each other because at the points of intersection, the y-values of both equations are the same.
So, we set 4x equal to 2x - 2 and solve for x.
4x = 2x - 2
By subtracting 2x from both sides, we get:
4x - 2x = -2
Which simplifies to:
2x = -2
Then, by dividing both sides by 2, we get:
x = -1
So, the x-coordinate of the point(s) where the two graphs intersect is -1.
Part B: To find the solution to 4x = 2x - 2, we can create a table with integer values of x between -3 and 3 and substitute them into the equation.
We start by substituting x = -3 into the equation:
4(-3) = 2(-3) - 2
-12 = -6 - 2
-12 = -8
Next, we substitute x = -2:
4(-2) = 2(-2) - 2
-8 = -4 - 2
-8 = -6
Continuing this process, we substitute x = -1:
4(-1) = 2(-1) - 2
-4 = -2 - 2
-4 = -4
Next, we substitute x = 0:
4(0) = 2(0) - 2
0 = 0 - 2
0 = -2
Continuing, we substitute x = 1:
4(1) = 2(1) - 2
4 = 2 - 2
4 = 0
Next, we substitute x = 2:
4(2) = 2(2) - 2
8 = 4 - 2
8 = 2
Finally, we substitute x = 3:
4(3) = 2(3) - 2
12 = 6 - 2
12 = 4
So, by creating this table, we can see that the solutions to the equation 4x = 2x - 2 are x = -1 and x = 2.
Part C: To solve the equation 4x = 2x - 2 graphically, we can plot the graphs of the two equations on a coordinate plane.
First, graph the equation y = 4x by plotting several points. For example, when x = -3, y = 4(-3) = -12, so plot the point (-3, -12). Similarly, when x = -2, y = 4(-2) = -8, so plot the point (-2, -8). Connect these plotted points to create the graph of y = 4x.
Next, graph the equation y = 2x - 2. Similarly, plot several points by substituting different values of x. For example, when x = -3, y = 2(-3) - 2 = -8, so plot the point (-3, -8). Similarly, when x = -2, y = 2(-2) - 2 = -6, so plot the point (-2, -6). Connect these plotted points to create the graph of y = 2x - 2.
The points where the two graphs intersect represent the solutions to the equation 4x = 2x - 2. By visually inspecting the graph, we can see that the points of intersection are approximately (-1, -4) and (2, 2). These are the solutions to the equation 4x = 2x - 2.
Part A: The x-coordinates of the points where the graphs of the equations y = 4x and y = 2x−2 intersect represent the values of x for which both equations have the same y-value. In other words, the points of intersection satisfy the condition that the y-values of both equations are equal. The equation 4x = 2x−2 is formed by setting the two expressions for y equal and solving for x. Therefore, the values of x that satisfy this equation will also be the x-coordinates of the points where the two graphs intersect.
Part B: To find the solution to the equation 4x = 2x−2, we can create a table and substitute 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
From the table, we can see that the solution to the equation 4x = 2x−2 is x = 1.
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 the same Cartesian coordinate system. The point(s) of intersection on the graph will represent the solution(s) to the equation.
1. Plot the graph of y = 4x: To do this, choose some x-values and calculate the corresponding y-values using the equation y = 4x. Connect the points to obtain a straight line.
2. Plot the graph of y = 2x−2: Similarly, choose some x-values and calculate the corresponding y-values using the equation y = 2x−2. Connect the points to obtain another straight line.
3. Locate the point(s) of intersection: The point(s) where the two lines intersect represent the solutions to the equation 4x = 2x−2. You can identify the x-coordinate(s) of the intersection point(s) by visual observation or by drawing a vertical line from the intersection point(s) to the x-axis and reading the value(s) of x.
Alternatively, you can also solve the equation graphically by using graphing software or online graphing tools, which will help you visualize the intersection point(s) more accurately.