Solve this simultaneous equation:
$\begin{array}{l}
5 x+3 y=9 \
7 x-4 y=14
\end{array}$
Answer (2)
Multiply equations to make the coefficients of y opposites.
Add the equations to eliminate y and solve for x : x = 41 78 .
Substitute the value of x into one of the original equations and solve for y : y = 41 − 7 .
The solution to the system of equations is x = 41 78 , y = 41 − 7 .
Explanation
Problem Analysis We are given a system of two linear equations with two variables, x and y :
5 x + 3 y = 9 7 x − 4 y = 14
Our objective is to solve this system of equations to find the values of x and y . We will use the method of elimination to solve for x and y .
Eliminating y Multiply the first equation by 4 and the second equation by 3 to eliminate y :
4 ( 5 x + 3 y ) = 4 ( 9 ) ⇒ 20 x + 12 y = 36
3 ( 7 x − 4 y ) = 3 ( 14 ) ⇒ 21 x − 12 y = 42
Solving for x Add the two resulting equations to eliminate y :
( 20 x + 12 y ) + ( 21 x − 12 y ) = 36 + 42
41 x = 78
Solve for x :
x = 41 78
Solving for y Substitute the value of x back into the first original equation to solve for y :
5 ( 41 78 ) + 3 y = 9
41 390 + 3 y = 9
3 y = 9 − 41 390
3 y = 41 369 − 41 390
3 y = 41 − 21
y = 41 × 3 − 21
y = 41 − 7
Final Answer Therefore, the solution to the system of equations is:
x = 41 78 , y = 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 products to maximize profit, or modeling supply and demand in economics. For example, a company might use a system of equations to determine the number of units they need to sell to cover their costs and start making a profit. Understanding how to solve systems of equations is crucial for making informed decisions in business and economics.
To solve the equations 5x + 3y = 9 and 7x - 4y = 14, we use the elimination method by aligning coefficients and eliminating y. This yields x = 78/41 and y = -7/41. The final solution is x ≈ 1.90 and y ≈ -0.17.
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