Solve the system of equations:
[tex]
\begin{array}{l}
x=3 y+1 \\
2 x+4 y=12
\end{array}
[/tex]
Answer (2)
Substitute x = 3 y + 1 into the second equation.
Simplify and solve for y : 2 ( 3 y + 1 ) + 4 y = 12 ⇒ y = 1 .
Substitute y = 1 back into the first equation to solve for x : x = 3 ( 1 ) + 1 ⇒ x = 4 .
The solution to the system of equations is ( 4 , 1 ) .
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y :
Equation 1: x = 3 y + 1 Equation 2: 2 x + 4 y = 12
Our goal is to find the values of x and y that satisfy both equations simultaneously.
Substitution We can use the substitution method to solve this system. Since Equation 1 already expresses x in terms of y , we can substitute this expression into Equation 2:
2 ( 3 y + 1 ) + 4 y = 12
Solve for y Now, we simplify and solve for y :
6 y + 2 + 4 y = 12 10 y + 2 = 12 10 y = 10 y = 1
Solve for x Now that we have the value of y , we can substitute it back into Equation 1 to find the value of x :
x = 3 ( 1 ) + 1 x = 3 + 1 x = 4
State the solution Therefore, the solution to the system of equations is x = 4 and y = 1 . We can write this as an ordered pair ( 4 , 1 ) .
Examples
Systems of equations are used in various real-life scenarios, such as determining the break-even point for a business. For example, if a company has fixed costs and variable costs, and they sell a product at a certain price, they can set up a system of equations to find the number of units they need to sell to cover their costs and start making a profit. Another example is in physics, where systems of equations can be used to analyze the forces acting on an object.
The solution to the system of equations is x = 4 and y = 1 . This can be expressed as the ordered pair ( 4 , 1 ) . Using substitution made it straightforward to find these values.
;
