A math student has a plan to solve the following system by the elimination method. To eliminate the $x$-terms, he wants to multiply the top equation by 7. What should he multiply the second equation by so that when he adds the equations, the $x$ terms are eliminated?
[tex]
\begin{array}{l}
-3 x-7 y=-56 \\
-7 x+10 y=1
\end{array}
[/tex]
Answer (2)
To eliminate the x terms when solving the given system of equations, multiply the first equation by 7 and the second equation by -3. This will result in the x terms cancelling out when the equations are added together. Therefore, multiply the second equation by − 3 .
;
Multiply the first equation by 7 to get − 21 x − 49 y = − 392 .
Determine the factor k to multiply the second equation by, such that the x terms cancel out when the equations are added.
Set up the equation − 21 − 7 k = 0 and solve for k , which gives k = − 3 .
Therefore, the second equation should be multiplied by − 3 .
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y :
− 3 x − 7 y = − 56 − 7 x + 10 y = 1
The goal is to eliminate the x terms by multiplying the top equation by 7 and finding the correct factor to multiply the second equation by.
Multiply the first equation by 7 First, multiply the first equation by 7:
7 ( − 3 x − 7 y ) = 7 ( − 56 ) − 21 x − 49 y = − 392
Multiply the second equation by k Let k be the factor we need to multiply the second equation by. Then we have:
k ( − 7 x + 10 y ) = k ( 1 ) − 7 k x + 10 k y = k
Find the value of k To eliminate the x terms when we add the modified equations, the coefficients of x must be opposites. That is, we want the x term in the second equation to be 21 x so that when we add the two equations, the x terms cancel out. Therefore, we need − 7 k = 21 , so k = − 3 . However, the problem states that we want to eliminate the x terms by multiplying the top equation by 7. This results in -21x. Therefore, we need to find k such that -7k is the opposite of -21, which is 21. Thus, -7k = 21, which means k = -3. However, the question asks what we should multiply the second equation by so that when he adds the equations, the x terms are eliminated. So we want -21x + (-7kx) = 0, which means -21 - 7k = 0, so -7k = 21, and k = -3.
Verify the result If we multiply the second equation by − 3 , we get:
− 3 ( − 7 x + 10 y ) = − 3 ( 1 ) 21 x − 30 y = − 3
Adding this to the modified first equation, we have:
( − 21 x − 49 y ) + ( 21 x − 30 y ) = − 392 + ( − 3 ) − 79 y = − 395
So the x terms are eliminated.
State the final answer Therefore, the second equation should be multiplied by − 3 .
Examples
When balancing chemical equations, you often need to multiply entire equations by certain factors to ensure that the number of atoms of each element is the same on both sides of the equation. This is analogous to the elimination method in solving systems of equations, where you multiply equations by constants to eliminate variables and find the solution.
