[tex]$\begin{array}{l}5 x+3 y=9 \ 7 x-4 y=14\end{array}$[/tex]
Answer (2)
Multiply the first equation by 4 and the second equation by 3 to prepare for eliminating y .
Add the modified equations to eliminate y and solve for x : x = 41 78 .
Substitute the value of x back into one of the original equations and solve for y : y = − 41 7 .
The solution to the system of equations is ( 41 78 , − 41 7 ) .
Explanation
Analyze the problem We are given a system of two linear equations with two variables, x and y . Our goal is to find the values of x and y that satisfy both equations simultaneously. The given equations are:
5 x + 3 y = 9 7 x − 4 y = 14
We will use the method of elimination to solve this system.
Eliminate y To eliminate y , we can multiply the first equation by 4 and the second equation by 3. This will make the coefficients of y in both equations equal in magnitude but opposite in sign.
Multiplying the first equation by 4, we get: 4 ( 5 x + 3 y ) = 4 ( 9 ) 20 x + 12 y = 36
Multiplying the second equation by 3, we get: 3 ( 7 x − 4 y ) = 3 ( 14 ) 21 x − 12 y = 42
Solve for x Now, we add the two resulting equations to eliminate y :
( 20 x + 12 y ) + ( 21 x − 12 y ) = 36 + 42 41 x = 78
Now, we solve for x :
x = 41 78
Solve for y Now that we have the value of x , we can substitute it back into either of the original equations to solve for y . Let's use the first equation: 5 x + 3 y = 9 5 ( 41 78 ) + 3 y = 9 41 390 + 3 y = 9
Subtract 41 390 from both sides: 3 y = 9 − 41 390 3 y = 41 369 − 41 390 3 y = 41 − 21
Divide by 3: y = 41 ⋅ 3 − 21 y = 41 − 7
State the solution Therefore, the solution to the system of equations is: x = 41 78 , y = 41 − 7
We can write the solution as an ordered pair: ( 41 78 , 41 − 7 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business, calculating the optimal mix of ingredients in a recipe, or modeling supply and demand in economics. For instance, suppose a company sells two products. By setting up a system of equations representing the cost and revenue for each product, one can find the quantity of each product that needs to be sold to break even. This helps in making informed business decisions and optimizing resource allocation. Let's say product A's equation is 2 x + 3 y = 10 and product B's equation is 4 x − y = 6 . Solving this system gives the quantities of A and B needed for a specific outcome.
To solve the equations 5 x + 3 y = 9 and 7 x − 4 y = 14 , we used elimination by multiplying the equations to eliminate y , solving for x , and substituting back to find y . The solution is x = 41 78 and y = 41 − 7 , expressed as the ordered pair igg(\frac{78}{41}, \frac{-7}{41}igg) .
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