Solve the system of equations and enter the solution as an ordered pair in the box below.
[tex]\begin{array}{l}
3 y=2 x+1 \\
5 y=2 x+3
\end{array}[/tex]
Solve the system of equations and enter the solution as an ordered pair in the box below.
[tex]\begin{array}{c}
5 x+3 y=11 \\
2 x-y=0
\end{array}[/tex]
Answer (2)
Solve the second equation for y : y = 2 x .
Substitute this expression into the first equation: 5 x + 3 ( 2 x ) = 11 .
Simplify and solve for x : x = 1 .
Substitute x = 1 back into y = 2 x to find y = 2 . The solution is ( 1 , 2 ) .
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. We will solve this system using the substitution method.
Solving for y in the Second Equation First, let's solve the second equation for y in terms of x :
2 x − y = 0 Add y to both sides: 2 x = y So, y = 2 x .
Substituting into the First Equation Now, substitute this expression for y into the first equation: 5 x + 3 y = 11 5 x + 3 ( 2 x ) = 11
Solving for x Simplify and solve for x :
5 x + 6 x = 11 11 x = 11 Divide both sides by 11: x = 1
Solving for y Now that we have the value of x , we can substitute it back into the equation y = 2 x to find the value of y :
y = 2 ( 1 ) y = 2
Final Answer Therefore, the solution to the system of equations is the ordered pair ( x , y ) = ( 1 , 2 ) .
Examples
Systems of equations are used in various real-world applications, such as determining the break-even point for a business. For example, suppose a company wants to know how many units of a product they need to sell to cover their costs. They can set up a system of equations where one equation represents the cost function and the other represents the revenue function. By solving the system, they can find the number of units they need to sell to break even.
The solutions to the systems of equations are ( 1 , 1 ) for the first system and ( 1 , 2 ) for the second system. Each system was solved using substitution and simplification of the equations. This method allows us to find the values of x and y that satisfy both equations simultaneously.
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