Solve the simultaneous equations:
$\begin{array}{l}
10 x+y=29 \
7 x+y=20
\end{array}$
Answer (2)
Subtract the second equation from the first to eliminate y : 3 x = 9 .
Solve for x : x = 3 .
Substitute x = 3 into the second equation: 7 ( 3 ) + y = 20 .
Solve for y : y = − 1 . The solution is x = 3 , y = − 1 .
Explanation
Understanding 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.
Setting up the Solution We have the following system of equations:
10 x + y = 29 7 x + y = 20
We can solve this system using the elimination method.
Solving for x Subtract the second equation from the first equation to eliminate y :
( 10 x + y ) − ( 7 x + y ) = 29 − 20 10 x − 7 x + y − y = 9 3 x = 9
Now, divide both sides by 3 to solve for x :
x = 3 9 x = 3
Solving for y Substitute the value of x into either of the original equations to solve for y . Let's use the second equation:
7 x + y = 20 7 ( 3 ) + y = 20 21 + y = 20
Subtract 21 from both sides to solve for y :
y = 20 − 21 y = − 1
Checking the Solution Now we have x = 3 and y = − 1 . Let's check our solution by substituting these values into both original equations:
Equation 1: 10 x + y = 29 10 ( 3 ) + ( − 1 ) = 30 − 1 = 29
Equation 2: 7 x + y = 20 7 ( 3 ) + ( − 1 ) = 21 − 1 = 20
Both equations are satisfied, so our solution is correct.
Final Answer The solution to the system of equations is x = 3 and y = − 1 .
Examples
Simultaneous equations are used in various fields, such as economics and physics. For example, in economics, you might have two equations representing the supply and demand curves for a product. Solving these equations simultaneously gives you the equilibrium price and quantity. In physics, you might use simultaneous equations to solve for the forces acting on an object in static equilibrium. Understanding how to solve these equations is crucial for making predictions and analyzing real-world scenarios.
The solution to the simultaneous equations is x = 3 and y = − 1 after applying the elimination method. First, we eliminated y by subtracting the second equation from the first, then solved for x and subsequently for y . Finally, we verified both values satisfy the original equations.
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