To answer this, you will need to set up two (2) linear equations. We know 2 things; the total value of $10 and $5 bills is$890 and that she has 130 bills in total.
Let's let "x" represent the number of $10 bills and "y" be the $5 bills,
The equations will look as follows:
x + y = 130 ~~the number of $10 bills "x" and $5 bills "y" is 130
10x + 5y = 890~~the total value of $10 bills "x" and $5 bills "y" is $890
when solving a system of linear equations as demonstrated above we can use a number of methods. The most in depth method, in my opinion, would be substitution. I recommend this method because it forces the student to manipulate the equations to find an effective method in solving it.
to use the substitution method, you will need to isolate one of the variables "x" or "y" do not feel pressured to always solve for "x" or vice versa. 'solvers preference'
So, taking the first equation; x + y = 130 'seemed to be the easiest to isolate' we get;
x + y = 130
- y - y ~~Subtract y from both sides to isolate the variable x; we get:
x = -y + 130 or x = 130 - y
since we now have isolated a variable, we can SUBSTITUTE into the other equation and this will allow us to solve for one variable and substitute that value into the equation to solve for the other variable.
Now we will substitute the equation x = 130 - y into the other equation. The equation will look as follows:
10x + 5y = 890
10(130 - y) + 5y = 890~~substitute 130 - y in for x.
1300-10y + 5y = 890~~apply the distributive property
1300 - 5y = 890~~isolate and solve for y
-1300 -1300
-5y = -410~~Divide by the coefficient-SIGN INCLUDED
-5 -5
y = 82
Now we have solved for 1 variable, we know that we have 82 $5 bills, but we are not done yet. Fortunately, the hard part is over. We know that from our original system of equations, we have a total of 130 bills and a total amount of $890. So, we are going to use this information to help us find out how many $10 bills we had. To do this, we are going to substitute our value for y; 82, which we just solved for.
x + y = 130
x + (82) = 130~~Substitute 82 in for y
-82 = -82~~Solve for X
x = 48
If, at any point when you are substituting, are unsure if you are on the right track check your answers. We solved for y earlier to get 82 and we just solved for x to get 48, the proposed answers should equal out to 130. Plug into a calculator and check. 48+82 does in fact equal 130. What about the other equation? If these are truly the right values, the proposed answers for both x and y MUST work in BOTH equations, so lets check that real quick
The second equation was 10x + 5y = 890 so lets plug in and evaluate x and y
10x + 5y = 890
10(48) + 5(82) = 890~~plug in x and y respectively
480 + 410 = 890~~Evaluate...
890 = 890~~890 does in fact equal 890 so the values for x and y are 48 and 82 respectively.
If, whoever reads this would like me to go over the other methods or reiterate on this one (as we all have different learning styles) please feel free to reach out to me, as I am always up for helping other students and colleagues alike.
Have a wonderful day!!
--Michael De La Cruz
