Solve the following simultaneous equations algebraically:
[tex]\begin{array}{l}
7 x+5 y=8 \
3 x-2 y=-9
\end{array}[/tex]
Answer (2)
Multiply the first equation by 2 and the second equation by 5 to prepare for eliminating y .
Add the modified equations to eliminate y and solve for x : x = − 1 .
Substitute the value of x into one of the original equations and solve for y : y = 3 .
The solution to the system of equations is x = − 1 , y = 3 .
Explanation
Analyze the problem and choose a method 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 equations are:
7 x + 5 y = 8 3 x − 2 y = − 9
We will use the elimination method to solve this system.
Multiply equations to prepare for elimination To eliminate y , we can multiply the first equation by 2 and the second equation by 5. This will make the coefficients of y in both equations equal in magnitude but opposite in sign.
Multiply the first equation by 2: 2 ( 7 x + 5 y ) = 2 ( 8 ) 14 x + 10 y = 16
Multiply the second equation by 5: 5 ( 3 x − 2 y ) = 5 ( − 9 ) 15 x − 10 y = − 45
Add the equations to eliminate y Now, add the two modified equations to eliminate y :
( 14 x + 10 y ) + ( 15 x − 10 y ) = 16 + ( − 45 ) 14 x + 15 x + 10 y − 10 y = 16 − 45 29 x = − 29
Solve for x Divide both sides by 29 to solve for x :
29 29 x = 29 − 29 x = − 1
Substitute x into the first equation Substitute the value of x into one of the original equations to solve for y . We'll use the first equation: 7 x + 5 y = 8 7 ( − 1 ) + 5 y = 8 − 7 + 5 y = 8
Isolate the term with y Add 7 to both sides: − 7 + 5 y + 7 = 8 + 7 5 y = 15
Solve for y Divide both sides by 5 to solve for y :
5 5 y = 5 15 y = 3
State the solution Therefore, the solution to the system of equations is x = − 1 and y = 3 .
We can write the solution as an ordered pair: ( − 1 , 3 ) .
Verify the solution To check our solution, we can substitute x = − 1 and y = 3 into both original equations:
First equation: 7 x + 5 y = 8 7 ( − 1 ) + 5 ( 3 ) = 8 − 7 + 15 = 8 8 = 8
Second equation: 3 x − 2 y = − 9 3 ( − 1 ) − 2 ( 3 ) = − 9 − 3 − 6 = − 9 − 9 = − 9
Since the solution satisfies both equations, it is correct.
Final Answer The solution to the system of equations is x = − 1 and y = 3 .
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 bakery sells cakes and pies. Each cake requires 2 hours of labor and 1 kg of flour, while each pie requires 1 hour of labor and 0.5 kg of flour. If the bakery has 20 hours of labor and 10 kg of flour available, we can set up a system of equations to determine how many cakes and pies the bakery can make. Solving this system helps the bakery optimize its production given its constraints. Let c be the number of cakes and p be the number of pies. Then we have the equations 2 c + p = 20 and c + 0.5 p = 10 . Solving this system gives the optimal number of cakes and pies to produce.
The solution to the given simultaneous equations is x = − 1 and y = 3 . This can be expressed in ordered pair form as ( − 1 , 3 ) . Using the elimination method, we manipulated and solved the equations step-by-step, confirming both values satisfy the original equations.
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