Can you explain how to find the solutions to the following system of equation problems?
Graph f(x) and g(x) to find the solution to the equation f(x) = g(x). Approximate the answer when necessary.
1. f(x) = -2/3x +4 and g(x)= x - 6
Ans:
Step1 graph lines. f(x) has y-int of 4 and slope of -2/3. g(x) has y-int of -6 and slope of 1.
Step2 look where lines intercept. point (6,0)
step3 check answer by substituting (6,0) into both f(x) and g(x)?
f(x): 0=-2/3(6) +4
0 = -4 +4
0 = 0 (solution)
g(x): 0 = 6 - 6
0 = 0
So, (6,0) is the solution to f(x) and g(x)
2. f(x) = -5/6x and g(x) = 2/5x + 4
Ans:
Step1 graph lines. f(x) has y-int of 0 and slope of -5/6. g(x) has y-int of 4 and slope of 2/5.
Step2 look where lines intercept. It's hard to tell. Estimated at (-3.1, 2.5).
step3 check answer by substituting (-3.1, 2.5) into both f(x) and g(x)?
f(x): 2.5 =-5/6(-3.1)
2.5 = -2.58 (not a solution?)
g(x): 2.5 = 2/5(-3.1)+ 4
2.5 = 1.24 + 4
2.5 = 5.24 (not a solution?)
So, (-3.1, 2.5) is not the solution to f(x) and g(x)?
#1, done well
#2, your sketch should look like this
http://www.wolframalpha.com/input/?i=plot+f(x)+%3D+-5%2F6x+,+g(x)+%3D+2%2F5x+%2B+4
look at the 2nd graph
looks like appr (-3.25, 2.5) to me, so your guess is close
your check:
f(x): 2.5 =-5/6(-3.1)
2.5 = -2.58 <----- should be +2.58, close enough
g(x): 2.5 = 2/5(-3.1)+ 4
2.5 = 1.24 + 4 <----- -1.24 + 4
2.5 = 2.76
had you used -3.25
2.5 = (2/5)(-3.25) + 4
2.5 = -1.3+4
2.5 = 2.7
To find the solutions to a system of equations, you can use the graphing method. Here's how to go about it:
1. Start by graphing the two equations, f(x) and g(x), on the same coordinate plane.
- For f(x) = -2/3x +4, you can start by plotting the y-intercept, which is 4, and then using the slope of -2/3 to find another point.
- For g(x) = x - 6, start by plotting the y-intercept, which is -6, and then use the slope of 1 to find another point.
2. Look for the point where the two graphs intersect. This point represents the x-value (and the corresponding y-value) where f(x) is equal to g(x).
3. To approximate the answer, you can read the approximate coordinates of the point of intersection from the graph.
4. To check if the point of intersection is indeed a solution, substitute the x and y values of the point into both equations and check if both equations are satisfied.
Let's apply these steps to the first set of equations:
1. For f(x) = -2/3x +4, plot the y-intercept (0, 4), and use the slope of -2/3 to find another point. For example, if x = 6, then y = -2/3(6) + 4 = 0. So, another point is (6, 0).
For g(x) = x - 6, plot the y-intercept (0, -6), and use the slope of 1 to find another point. For example, if x = 6, then y = 6 - 6 = 0. So, another point is (6, 0).
2. The two lines intersect at the point (6, 0).
3. Substitute (6, 0) into both f(x) and g(x):
- For f(x): 0 = -2/3(6) + 4
Simplifying, 0 = -4 + 4, which is true.
- For g(x): 0 = 6 - 6, which is also true.
So, (6, 0) is the solution to the system of equations.
Now let's apply the same steps to the second set of equations:
1. For f(x) = -5/6x, plot the y-intercept (0, 0), and use the slope of -5/6 to find another point.
- For g(x) = 2/5x + 4, plot the y-intercept (0, 4), and use the slope of 2/5 to find another point.
2. The two lines do not appear to intersect at an exact point. However, you can estimate the coordinates of the point of intersection. In this case, the estimated point is (-3.1, 2.5).
3. Substitute (-3.1, 2.5) into both f(x) and g(x):
- For f(x): 2.5 = -5/6(-3.1)
Simplifying, 2.5 ≈ -2.58 (not a solution).
- For g(x): 2.5 = 2/5(-3.1) + 4
Simplifying, 2.5 ≈ 1.24 + 4 ≈ 5.24 (not a solution).
Therefore, (-3.1, 2.5) is not the solution to the system of equations.
Remember, when using the graphing method, there may be cases where the lines do not intersect or intersect at a point that is difficult to pinpoint accurately. In such cases, estimating the approximate coordinates from the graph and then checking the solutions by substitution is a way to proceed.