- A CHEMIST NEEDS TO MAKE TWO LITERS OF A 15% ACID SOLUTION
FROM HER 10% ACID SOLUTION AND A 35% ACID SOLUTION.
SO WE'RE MIXING THE 10% AND 35% SOLUTION
TO CREATE TWO LITERS OF 15% SOLUTION.
WE WANT TO KNOW HOW MANY LITERS OF EACH SHOULD SHE MIX
TO GET THE DESIRED SOLUTION.
SO WE'RE GOING TO NEED TWO VARIABLES HERE
AS WELL AS TWO EQUATIONS.
LET'S LET X EQUAL THE NUMBER OF LITERS OF THE 10% SOLUTION.
AND WE'LL LET Y EQUAL THE NUMBER OF NUMBER OF LITERS
OF THE 35% SOLUTION.
NOW SEE IF WE CAN WRITE OUR TWO EQUATIONS.
WE KNOW AFTER MIXING THESE TWO AMOUNTS,
WE'LL HAVE EXACTLY TWO LITERS OF THE 15% SOLUTION.
SO OUR QUANTITY EQUATION WILL BE X + Y MUST EQUAL TWO LITERS.
THE SECOND EQUATION
IS GOING DEAL WITH THE CONCENTRATIONS OF EACH SOLUTION.
SINCE X IS EQUAL TO THE NUMBER OF LITERS OF THE 10% SOLUTION
AND Y IS EQUAL TO THE NUMBER OF LITERS OF THE 35% SOLUTION,
10% x X, OR 0.10 x X + 35% x Y, OR 0.35 x Y MUST EQUAL 15%
OR 0.15 x THE TOTAL AMOUNT OF THE SOLUTION,
WHICH IS 2 LITERS.
SO HERE'S OUR SYSTEM OF EQUATIONS THAT WE HAVE TO SOLVE.
THE FORM OF THE FIRST EQUATION WILL BE GOOD TO WORK WITH,
BUT LET'S TRANSFORM THE SECOND EQUATION
BEFORE WE SOLVE THE SYSTEM.
LET'S START BY ELIMINATING THE DECIMALS.
IF WE MULTIPLY EVERYTHING BY 100,
THAT'S THE SAME AS MOVING THE DECIMAL POINT
TO THE RIGHT TWO PLACES.
SO WE CAN WRITE THIS AS 10X + 35Y EQUALS--
AND THIS WOULD BE 15 x 2, WHICH IS EQUAL TO 30.
SO NOW WE'LL SOLVE THE SYSTEM OF EQUATIONS
USING THESE TWO EQUATIONS.
SO AGAIN WE HAVE X + Y = 2 AND THEN WE HAVE 10X + 35Y = 30.
NOW WE CAN SOLVE THIS ALGEBRAICALLY
USING ELIMINATION OR SUBSTITUTION.
IF WE CALL THIS EQUATION 1 AND THIS EQUATION 2,
WE COULD VERY EASILY SOLVE THE FIRST EQUATION FOR X OR Y.
TO SOLVE THIS FOR X, WE WOULD SUBTRACT Y ON BOTH SIDES.
SO USING EQUATION 1, WE WOULD HAVE X = 2 - Y.
AND THEN WE CAN PERFORM SUBSTITUTION INTO EQUATION 2.
IF X = 2 - Y, WE CAN REPLACE X WITH 2 - Y.
NOW EQUATION 2 AFTER SUBSTITUTION,
WOULD BE 10 x 2 - Y, INSTEAD OF TIMES X, + 35Y = 30.
AND NOW WE HAVE ONE EQUATION WITH ONE VARIABLE,
SO NOW WE'LL SOLVE FOR Y.
WE HAVE 20 - 10Y + 35Y = 30.
WELL HERE WE HAVE 2Y TERMS SO WE CAN COMBINE THESE TWO TERMS.
-10Y + 35Y = 25Y, SO WE'LL HAVE 20 + 25Y = 30.
SUBTRACT 20 ON BOTH SIDES WOULD GIVE US 25Y = 10.
DIVIDE BOTH SIDES BY 25.
AND WE HAVE Y = 10/25 WHICH DOES SIMPLIFY.
THERE'S A COMMON FACTOR OF 5 HERE. SO WE HAVE Y = 2/5.
REMEMBER Y IS THE NUMBER OF LITERS OF THE 35% SOLUTION.
SO AS AN ORDERED PAIR WE KNOW THAT Y = 2/5
WE STILL HAVE TO GO BACK AND DETERMINE X,
BUT WE DO KNOW FROM BEFORE X = (2 - Y).
SO WE'D HAVE 2 - 2/5, WHICH IS 1 3/5.
SO WE HAVE 1 3/5 LITERS OF THE 10% SOLUTION.
WE MAY WANT TO CONVERT THESE TO DECIMALS.
3/5 IS THE SAME AS 0.6.
SO THE MIXTURE CONTAINS 1.6 LITERS OF THE 10% SOLUTION
AND 2/5 AS A DECIMAL WOULD BE 0.4 LITERS OF THE 35% SOLUTION.
OKAY. I HOPE YOU FOUND THIS HELPFUL.