Solve the system of equations:
[tex]
\begin{array}{l}
y=6 x+13 \\
y=2 x+1
\end{array}
[/tex]
A. (5,3)
B. (-3,-5)
C. (3,-5)
D. No solution
Answer (2)
Set the two equations equal to each other: 6 x + 13 = 2 x + 1 .
Solve for x : 4 x = − 12 , so x = − 3 .
Substitute x = − 3 into one of the equations to solve for y : y = 2 ( − 3 ) + 1 = − 5 .
The solution to the system of equations is ( − 3 , − 5 ) .
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. The equations are:
Equation 1: y = 6 x + 13 Equation 2: y = 2 x + 1
Setting the Equations Equal Since both equations are solved for y , we can set the expressions for y equal to each other:
6 x + 13 = 2 x + 1
Solving for x Now, we solve the resulting equation for x . Subtract 2 x from both sides:
6 x − 2 x + 13 = 2 x − 2 x + 1
4 x + 13 = 1
Subtract 13 from both sides:
4 x + 13 − 13 = 1 − 13
4 x = − 12
Divide both sides by 4:
4 4 x = 4 − 12
x = − 3
Solving for y Substitute the value of x back into either equation to solve for y . Let's use Equation 2:
y = 2 x + 1
y = 2 ( − 3 ) + 1
y = − 6 + 1
y = − 5
The Solution So, the solution is x = − 3 and y = − 5 . We write the solution as an ordered pair ( x , y ) = ( − 3 , − 5 ) .
Checking the Solution To check the solution, we substitute the values of x and y into both original equations to ensure they are satisfied.
Equation 1: y = 6 x + 13
− 5 = 6 ( − 3 ) + 13
− 5 = − 18 + 13
− 5 = − 5 (True)
Equation 2: y = 2 x + 1
− 5 = 2 ( − 3 ) + 1
− 5 = − 6 + 1
− 5 = − 5 (True)
Since the solution satisfies both equations, it is correct.
Final Answer The solution to the system of equations is ( − 3 , − 5 ) . Therefore, the correct answer is option b.
Examples
Systems of equations are used in various real-world applications. For example, they can be used to model supply and demand in economics, where one equation represents the supply curve and the other represents the demand curve. The solution to the system gives the equilibrium price and quantity. Another example is in physics, where systems of equations can be used to describe the motion of objects under certain constraints. Solving these systems helps in predicting the behavior of the objects.
The solution for the given system of equations is (-3, -5), which is option B. This solution was found by setting the equations equal, solving for x, and then substituting back to find y. Verification of both equations confirms the accuracy of the solution.
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