Translate the problem into a pair of linear equations in two variables. Solve the equations using either elimination or substitution. State your answer for the specified variable.
A student took out two loans totaling $10,000 to help pay for college expenses. One loan was at 8% simple interest, and the other was at 10%. After one year, the student owed $840 in interest. Find the amount of the loan at 10%.
x at .1 and y at .08
y = 10,000-x
.1x + .08y = 840
so substitute
.1x + .08 (10,000-x) = 840
x + y = 10000
.1x + .08y = 840
suggestion:
multiply the second by 100
then the first by 8 and subtract them
Thanks!! You are all the best.
To translate the problem into a pair of linear equations in two variables, we can assign variables to represent the unknowns. Let's denote the amount of the loan at 8% as "x" and the amount of the loan at 10% as "y."
The first equation can be derived from the fact that the total amount of the loans is $10,000:
x + y = 10,000 (Equation 1)
The second equation can be obtained by calculating the interest paid on each loan and adding them together. The interest on the 8% loan is 8% of x, which can be expressed as 0.08x. Similarly, the interest on the 10% loan is 10% of y, or 0.10y. Since the total interest paid is $840, we have:
0.08x + 0.10y = 840 (Equation 2)
Now that we have our system of equations, we can solve them using either elimination or substitution.
Let's use the substitution method. Solve Equation 1 for x:
x = 10,000 - y
Substitute this value of x into Equation 2:
0.08(10,000 - y) + 0.10y = 840
Now, simplify and solve for y:
800 - 0.08y + 0.10y = 840
0.02y = 40
y = 40/0.02
y = 2,000
Therefore, the amount of the loan at 10% is $2,000.