Video Transcript
Which of the following ordered pairs 𝑗, 𝑘 is the solution to the system of equations, three plus two-thirds 𝑘 equals five-sixths 𝑗, negative 28 minus six 𝑗 equals 20𝑘? Is it A) two, negative two, B) three, six, C) negative two, two, or D) six, three?
We could answer this problem by solving the two equations simultaneously, either by elimination or by substitution. Alternatively, we could substitute all four of the points into both of the equations. We will firstly look at solving the equations simultaneously. Let’s firstly consider the equation three plus two-thirds 𝑘 equals five sixths 𝑗 and call this equation one. We can eliminate the fractions from this equation by multiplying all three terms by six. Three multiplied by six is equal to 18. Two-thirds multiplied by six is equal to twelve-thirds. And 12 divided by three is equal to four. This means that six multiplied by two-thirds 𝑘 is equal to four 𝑘. Five-sixths 𝑗 multiplied by six is equal to five 𝑗 as the sixes cancel. Equation one therefore simplifies to 18 plus four 𝑘 is equal to five 𝑗.
Our 𝑘 term in equation two is 20𝑘. And multiplying four 𝑘 by five would give us 20𝑘. This means that we can multiply both sides of the new equation one by five. Five multiplied by 18 is equal to 90. As five multiplied by 10 is 50, and five multiplied by eight is 40. Four 𝑘 multiplied by five is equal to 20𝑘 and five 𝑗 multiplied by five is equal to 25𝑗.
We now have a 20𝑘 term in both equations. If we subtract 90 from both sides of the new equation, we can make 20𝑘 the subject. 20𝑘 is equal to 25𝑗 minus 90. We now know that 20𝑘 is equal to negative 28 minus six 𝑗 and 25𝑗 minus 90. This means that these two expressions must be equal to each other. 25𝑗 minus 90 is equal to negative 28 minus six 𝑗. Adding 90 to both sides of this equation gives us 25𝑗 is equal to 62 minus six 𝑗. As negative 28 plus 90 equals 62. Adding six 𝑗 to both sides of this new equation gives us 31𝑗 is equal to 62. Finally, dividing both sides by 31 gives us a value for 𝑗 equal to two. As 62 divided by 31 is two.
We can see from our four options that only one of them had a 𝑗-coordinate of two. This was option A. Whilst it looks like this is the correct answer, it is worth substituting it back into one of the equations to work out the value of 𝑘. Substituting 𝑗 equals two into the equation 20𝑘 is equal to 25𝑗 minus 90 gives us 20𝑘 is equal to 25 multiplied by two minus 90. 25 multiplied by two is equal to 50. Subtracting 90 from this gives us negative 40. This gives us 20𝑘 is equal to negative 40. And we can divide both sides by 20. Negative 40 divided by 20 is equal to negative two. Therefore, our value for 𝑘 is negative two. This is also the correct value of 𝑘 in option A. The ordered pair that is a solution to the system of equations is two, negative two, where 𝑗 equals two and 𝑘 equals negative two.
As mentioned at the start, we could have substituted the ordered pairs into the initial equations. Substituting in 𝑗 equals two and 𝑘 equals negative two into the first equation gives us three plus two-thirds multiplied by negative two is equal to five-sixths multiplied by two. Two-thirds multiplied by negative two is equal to negative four-thirds. Three plus negative four-thirds is the same as three minus four-thirds. Three is equal to nine-thirds. Therefore, we need to subtract four-thirds from nine-thirds. This is equal to five-thirds.
On the right-hand side, we need to multiply five-sixths by two. This is also equal to five-thirds. The ordered pair two, negative two is a solution to the first equation. Substituting the values into the second equation gives us negative 28 minus six multiplied by two is equal to 20 multiplied by negative two. Six multiplied by two is equal to 12. And 20 multiplied by negative two is equal to negative 40. Negative 28 minus 12 is equal to negative 40. So once again, the left-hand side is equal to the right-hand side. The ordered pair two, negative two is also a solution of the second equation. This confirms that the correct answer was option A.
