Solve this system of equations using the substitution method:
[tex]
\begin{array}{l}
y=2 x+4 \
4 x-6=y \
([?],
\end{array}
[/tex]
Answer (2)
Substitute y = 2 x + 4 into the second equation: 4 x − 6 = 2 x + 4 .
Solve for x : 2 x = 10 , so x = 5 .
Substitute x = 5 into y = 2 x + 4 to find y : y = 2 ( 5 ) + 4 = 14 .
The solution to the system of equations is ( 5 , 14 ) .
Explanation
Analyze the problem We are given a system of two equations with two variables, x and y . Our goal is to solve for x and y using the substitution method. The given equations are:
Equation 1: y = 2 x + 4 Equation 2: 4 x − 6 = y
Substitution Since we are using the substitution method, we can substitute the expression for y from Equation 1 into Equation 2. This means we replace y in the second equation with ( 2 x + 4 ) .
So, we have: 4 x − 6 = 2 x + 4
Solve for x Now, we solve the resulting equation for x . First, we subtract 2 x from both sides of the equation:
4 x − 2 x − 6 = 2 x − 2 x + 4 2 x − 6 = 4
Next, we add 6 to both sides of the equation:
2 x − 6 + 6 = 4 + 6 2 x = 10
Finally, we divide both sides by 2:
2 2 x = 2 10 x = 5
Solve for y Now that we have the value of x , we can substitute it back into either Equation 1 or Equation 2 to find the value of y . Let's use Equation 1:
y = 2 x + 4 y = 2 ( 5 ) + 4 y = 10 + 4 y = 14
State the solution So, the solution to the system of equations is x = 5 and y = 14 . We can write this as an ordered pair ( 5 , 14 ) .
Verify the solution To verify the solution, we substitute x = 5 and y = 14 into both equations:
Equation 1: 14 = 2 ( 5 ) + 4 ⇒ 14 = 10 + 4 ⇒ 14 = 14 (True) Equation 2: 4 ( 5 ) − 6 = 14 ⇒ 20 − 6 = 14 ⇒ 14 = 14 (True)
Since the solution satisfies both equations, it is correct.
Final Answer Therefore, the solution to the system of equations is ( x , y ) = ( 5 , 14 ) .
Examples
Systems of equations are used in various real-world applications. For instance, they can be used to model supply and demand in economics, where the intersection of the supply and demand curves represents the equilibrium price and quantity. They are also used in physics to solve problems involving multiple forces or constraints. In computer graphics, systems of equations are used to perform transformations and projections of 3D objects onto a 2D screen. Understanding how to solve systems of equations is therefore a valuable skill in many fields.
The solution to the system of equations using the substitution method is found to be ( 5 , 14 ) . By substituting y = 2 x + 4 into the second equation and solving, we find both values. Verification shows that these values satisfy both original equations.
;
