The first step to solving a system of equations using substitution is to substitute an equation equal to one variable in for that variable in the other equation.
2x+3(x+1.5)=12
The first step to solving a story problem is identifying variables. For this problem, I will identify the variables as:
D = Drew's age
J = Jimmy's age
Once the variables are identified, we need to find as many equations as we have variables. Since we have two variables, we will have to write two equations.
Since Drew is 3 years younger than Jimmy: D + 3 = J
Since the sum of the brothers' ages is 21: D + J = 21
Once I have two equations is two variables, I can solve the system using either substitution method or elimination method. For this problem, I will use substitution method.
D + J = 21 Equation 2
D + (D + 3) = 21 Substitution of value of J from equation 1 into equation 2
D + D + 3 = 21 Associative property of addition
2D + 3 = 21 Simplify
2D + 3 - 3 = 21 - 3 Subtract 3 from each side
2D = 18 Simplify each side
2D/2 = 18/2 Divide each side by 2
D = 9 Simplify each side
Now that we have D, we can substitute it into equation 1 to get the value of J
D + 3 = J Equation 1
9 + 3 = J Substitution
12 = J Simplify
- Make an equation representing the number of vehicles needed.
We have six drivers so
x + y ≤ 6
That's not really an equation; it's an inequality. We want to use all our drivers so we can use the small vans, so
x + y = 6
- Make an equation representing the total number of seats in vehicles for the orchestra members.
s = 25x + 12y
That's how many seats total; it has to be at least 111 so again an inequality,
25x + 12y ≥ 111
We solve it like a system of equations.
x + y = 6
y = 6 - x
111 = 25x + 12y = 25x + 12(6-x)
111 = 25x + 72 - 12x
111 - 72 = 13 x
39 = 13 x
x = 3
Look at that, it worked out exactly. It didn't have to.
y = 6 - x = 3
Answer: 3 buses, 3 vans
Visualize the distance traveled as the two side lengths of a right triangle, for this we can use the pythagorean theorem to solve for the hypotenuse (which is the shortest distance from his house to his office).
a²+b²=c²
6²+4²=c²
36+16=c²
52=c²
7.21=c
Answer=7.21km