System of Equations:
f(t)=7t+22 g(t)=10t+4 Solution of the system: (6,64) Explanation: See solution.
Practice makes perfect
We have been given a graph that shows the amount of water over time for two different functions.
We can write an equation for each line in slope-intercept form.
y=mt+ b
In this form, m is the slope and b is the y-intercept. We will find the slope of the lines by using the Slope Formula.
m=y_2-y_1/t_2-t_1
In the above formula, (t_1,y_1) and (t_2,y_2) represent two points of the line.
Equation for f(t)
Let's start with y=f(t), where t represents the number of days. Since the line passes through the points (0,22) and (6,64), we will substitute them into the Slope Formula in order to find the slope.
The slope of this line is 7.
f(t)=7t+ b
Next, we will determine the y-intercept. Since the line passes through the point (0, 22), we can immediately determine the y-intercept as 22. Thus, we can write the full equation of the line.
f(t)=7t+ 22
Equation for g(t)
We will write an equation for y=g(t) in the same way as we did for f(t). Since the line passes through the points (0,4) and (6,64), we will substitute them into the Slope Formula to find the slope.
Thus, the slope for this line is 10.
g(t)=10t+ b
Let's find the y-intercept now. Since the line passes through the point (0, 4), we can immediately determine the y-intercept as 4.
g(t)=10t+ 4
System of Equations
Finally, we have our system of linear equations.
f(t)=7t+22 & (I) g(t)=10t+4 & (II)
The points (0,22) and (0,4) indicate the initial amount of water, 22gal and 4gal. The solution of the system is the point of intersection of the lines, (6,64). It represents the amount of water after 6 minutes, which is 64gal.
