A one pound mix consisting of 30% cashews and 70% pistachios sells for $6.25. A one pound mix consisting of 80% cashews and 20% pistachios sells for $7.50. How would a mix consisting of 50% of each type of nut sell for?
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Khan Academy: Solving Systems of Equations by Multiplication
Guidance
In the linear systems in this lesson, we will need to multiply both equations by a constant in order to have opposite coefficients of one of the variables. In order to determine what numbers to multiply by, we will be finding the least common multiple of the given coefficients. Recall that the least common multiple of two numbers is the smallest number which is divisible by both of the given numbers. For example, 12 is the least common multiple of 4 and 6 because it is the smallest number divisible by both 4 and 6.
Example A
Solve the system using linear combinations:
Solution:
In this problem we cannot simply multiply one equation by a constant to get opposite coefficients for one of the variables. Here we will need to identify the least common multiple of the coefficients of one of the variables and use this value to determine what to multiple each equation by. If we look at the coefficients of
Now find
Solution: (5, 1)
Check your answer:
Example B
Solve the system using linear combinations:
Solution:
The first step is to decide which variable to eliminate. Either one can be eliminated but sometimes it is helpful to look at what we need to multiply by to eliminate each one and determine which is easier to eliminate. In general, it is easier to work with smaller numbers so in this case it makes sense to eliminate
Now find
Solution: (7, 2)
Check your answer:
Example C
Solve the system using linear combinations:
Solution:
This time, we will eliminate
Now find
Solution:
Check your answer:
Intro Problem Revisit
Write a system of linear equations to represent the given information. Let
Solve this system to determine the cost of each type of nut per pound. If we eliminate
Now find
So, we have determined that the cost of the cashews is $8 per pound and the cost of the pistachios is $5.50 per pound. Now we can determine the cost of the 50% mix as follows:
Guided Practice
Solve the following systems using linear combinations:
1.
2.
3.
Answers
1. We can eliminate either variable here. To eliminate
Now find
Solution: (8, 9)
2. Again we can eliminate either variable. To eliminate
Now find
Solution: (6, 5)
3. To start this one we need to get the second equation in standard form. The resulting system will be:
This time we just need to multiply the second equation by 3 to eliminate
Solution: There are infinite solutions.
Explore More
Solve the systems using linear combinations.

17x−5y2x+7y=4=46 
9x+2y11x+5y=−13=2 
3x+4y5x+5y=−16=−5 
2x−8y3x+7y=−8=45 
5x−10y6x+3y=60=−33 
3x+10y−5x−7y=−50=6 
11x+6y13x−5y=30=−25 
15x+2y18x−9y=23=−18 
12x+8y17x−12y=64=9 
11x−3y33x−36=12=9y 
4x+3y6x−13y=0=35 
18x+2y−12x−3y=−2=−7 
−6x+11y8x−15y=−109=149 
8x−32x+20y=−5y−1=8 
10x−16y−15x+14y=−12=−27
Set up and solve a system of equations to answer the following questions.
 A mix of 35% almonds and 65% peanuts sells for $5.70. A mix of 75% almonds and 25% peanuts sells for $6.50. How much should a mix of 60% almonds and 40% peanuts sell for?
 The Robinson family pays $19.75 at the movie theater for 3 medium popcorns and 4 medium drinks. The Jamison family pays $33.50 at the same theater for 5 medium popcorns and 7 medium drinks. How much would it cost for a couple to get 2 medium drinks and 2 medium popcorns?
 A cell phone company charges extra when users exceed their included call time and text message limits. One user paid $3.24 extra having talked for 240 extra minutes and sending 12 additional texts. A second user talked for 120 extra minutes and sent 150 additional texts and was charged $4.50 above the regular fee. How much extra would a user be charged for talking 140 extra minutes and sending 200 additional texts?