Solve this system of equations using the substitution method:
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\begin{array}{l}
y=2 x+9 \
8 x-3=y \
([?],
\end{array}
[/tex]
Answer (2)
Substitute the expression for y from the first equation into the second equation: 8 x − 3 = 2 x + 9 .
Solve for x : 6 x = 12 , so x = 2 .
Substitute the value of x back into the first equation to find y : y = 2 ( 2 ) + 9 = 13 .
The solution to the system of equations is ( 2 , 13 ) , so ( 2 , 13 ) .
Explanation
Understanding the Problem We are given a system of two equations:
Equation 1: y = 2 x + 9 Equation 2: 8 x − 3 = y
We need to find the values of x and y that satisfy both equations using the substitution method.
Substitution Since both equations are already solved for y , we can substitute the expression for y from Equation 1 into Equation 2. This means we replace y in Equation 2 with ( 2 x + 9 ) .
So, we have:
8 x − 3 = 2 x + 9
Solving for x Now, we solve the equation for x :
Subtract 2 x from both sides:
8 x − 2 x − 3 = 2 x − 2 x + 9
6 x − 3 = 9
Add 3 to both sides:
6 x − 3 + 3 = 9 + 3
6 x = 12
Divide both sides by 6 :
6 6 x = 6 12
x = 2
Solving 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 + 9
Substitute x = 2 :
y = 2 ( 2 ) + 9
y = 4 + 9
y = 13
Final Answer So, the solution to the system of equations is x = 2 and y = 13 . We can write this as an ordered pair ( x , y ) = ( 2 , 13 ) .
Examples
Systems of equations are used in various real-world applications. For example, when planning a wedding, you might have a budget and need to determine the number of guests you can invite while staying within your budget. If you have two different vendors with different pricing structures, you can set up a system of equations to find the number of guests that minimizes your costs. Another example is in electrical engineering, where systems of equations are used to analyze circuits and determine the currents and voltages at different points in the circuit. These systems help engineers design efficient and reliable electrical systems.
By using the substitution method, we solved the system of equations and found that the solution is (2, 13). This was achieved by equating the two expressions for y, solving for x, and then finding y. The final answer is an ordered pair: (2, 13).
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