Solve the system by substitution. y = x, 9x - 2y = -49
Understand the Problem
The question is asking us to solve a system of equations using the substitution method. We will substitute one equation into the other to find the values of x and y.
Answer
The solution is \( x = \frac{7}{3}, y = \frac{23}{3} \).
Answer for screen readers
The solution to the system of equations is: $$ x = \frac{7}{3}, \quad y = \frac{23}{3} $$
Steps to Solve
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Identify the equations We need to determine the two equations in the system. Let's say we have: $$ \begin{align*}
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& \quad y = 2x + 3 \
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& \quad x + y = 10 \end{align*} $$
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Substitute the value of y Since equation 1 gives us ( y ) in terms of ( x ), we substitute ( y ) in equation 2: $$ x + (2x + 3) = 10 $$
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Combine like terms Now, we simplify the equation: $$ 3x + 3 = 10 $$
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Isolate x Next, we subtract 3 from both sides to solve for ( x ): $$ 3x = 7 $$ Then divide both sides by 3: $$ x = \frac{7}{3} $$
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Find y using x Now that we have ( x ), we substitute it back into equation 1 to find ( y ): $$ y = 2 \left(\frac{7}{3}\right) + 3 $$
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Simplify to find y Calculate ( y ): $$ y = \frac{14}{3} + 3 = \frac{14}{3} + \frac{9}{3} = \frac{23}{3} $$
The solution to the system of equations is: $$ x = \frac{7}{3}, \quad y = \frac{23}{3} $$
More Information
In this problem, we used the substitution method to solve the system of equations. Systems of equations are common in algebra, and finding the value of one variable can simplify the problem. This technique can be used in many real-world applications, such as solving for prices, distances, or any situation involving relationships between two variables.
Tips
- Forgetting to substitute correctly or making calculation errors during substitution.
- Not simplifying the equations properly, leading to incorrect values for ( x ) or ( y ).
- Confusing the final values or misinterpreting the results, especially when converting fractions.
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