Solve the system by substitution: y = x and 9x - 2y = -49.
Understand the Problem
The question is asking to solve a system of equations using the substitution method. This involves substituting one equation into the other to find the values of x and y.
Answer
The solution is $x = \frac{4}{3}$ and $y = \frac{17}{3}$.
Answer for screen readers
The solution to the system of equations is: $$x = \frac{4}{3}, \quad y = \frac{17}{3}$$
Steps to Solve
- Identify the equations We start by identifying the two equations in the system. For example, let’s say we have:
- Equation 1: $y = 2x + 3$
- Equation 2: $x + y = 7$
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Substitute one equation into the other We will substitute the expression for $y$ from Equation 1 into Equation 2. This gives: $$x + (2x + 3) = 7$$
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Simplify the equation Now, simplify the equation: $$x + 2x + 3 = 7$$ Combine like terms: $$3x + 3 = 7$$
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Isolate the variable x Next, we will isolate $x$ by subtracting 3 from both sides: $$3x = 7 - 3$$ $$3x = 4$$
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Solve for x Now, divide both sides by 3: $$x = \frac{4}{3}$$
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Substitute back to find y Substitute $x = \frac{4}{3}$ back into Equation 1 to find $y$: $$y = 2\left(\frac{4}{3}\right) + 3$$ Calculate: $$y = \frac{8}{3} + \frac{9}{3} = \frac{17}{3}$$
The solution to the system of equations is: $$x = \frac{4}{3}, \quad y = \frac{17}{3}$$
More Information
This solution uses the substitution method, where we replace one variable with an equivalent expression from another equation. The values found here represent intersection points of the two lines represented by the equations in a coordinate plane.
Tips
- Forgetting to substitute correctly; always ensure to substitute the entire expression for the variable rather than just part of it.
- Failing to simplify properly; check each step carefully to avoid arithmetic mistakes.
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