To solve a system of two linear equations using substitution, follow these steps:
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Identify the method: The substitution method is often the best choice when one equation is already solved for one variable or can be easily arranged to do so.
Answer: Substitution -
Isolate a variable: Choose one equation and solve for one of the variables (usually \(x\) or \(y\)).
Answer: For example, if you have equations like \(y = 2x + 3\) and \(3x + 4y = 12\), you can isolate \(y\) in the first equation. -
Substitute that variable's expression: Take the expression for the isolated variable and substitute it into the other equation.
Answer: Substitute \(y = 2x + 3\) into the second equation \(3x + 4(2x + 3) = 12\). -
Solve the one-variable equation: Simplify and solve the resulting equation.
Answer: \(3x + 8x + 12 = 12\), then combine like terms to get \(11x + 12 = 12\), leading to \(11x = 0\), thus \(x = 0\). -
Substitute the value of that variable: Now that you have a value for one variable, substitute it back into the equation you used to isolate the variable.
Answer: Substitute \(x = 0\) back into \(y = 2(0) + 3\) to find \(y\). -
Solve for the second variable: This will give you the value of the second variable.
Answer: So, \(y = 3\). -
Check your answer: Substitute both values into the original equations to ensure they hold true. Alternatively, you can graph both equations to see where they intersect.
Answer: Check by plugging \(x = 0\) and \(y = 3\) into both original equations to confirm they satisfy both, verifying your solution.
So, the solution to the system of equations is \( (x, y) = (0, 3) \).
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