how to solve a set of equations with symbols such as X and Y and two equations

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Welcome to solving systems of equations! A system of equations is a collection of two or more equations that share the same variables. Our goal is to find the values of these variables that satisfy all equations simultaneously. For example, we have two equations: two x plus three y equals twelve, and x minus y equals one. The solution is the point where both equations are true at the same time. The substitution method is one of the most common ways to solve systems of equations. First, we solve one equation for one variable. From x minus y equals one, we get x equals y plus one. Next, we substitute this expression into the other equation. Two times y plus one, plus three y equals twelve. Expanding gives us two y plus two plus three y equals twelve. Combining like terms: five y plus two equals twelve. Solving for y: five y equals ten, so y equals two. Finally, we substitute back to find x: x equals two plus one, which equals three. Our solution is x equals three, y equals two. The elimination method works by eliminating one variable through addition or subtraction. We start with our original equations: two x plus three y equals twelve, and x minus y equals one. To eliminate y, we need equal coefficients. We multiply the second equation by three: three times x minus y equals three times one, giving us three x minus three y equals three. Now we have two x plus three y equals twelve, and three x minus three y equals three. Adding these equations eliminates y: five x equals fifteen, so x equals three. Substituting back into the second original equation: three minus y equals one, so y equals two. Our solution is x equals three, y equals two. Always verify your solution by substituting the values back into the original equations. Let's check our solution x equals three, y equals two. For the first equation: two times three plus three times two equals six plus six equals twelve. Check! For the second equation: three minus two equals one. Check! Both equations are satisfied. Here's a practice example for you to try: x plus two y equals eight, and three x minus y equals five. Remember to choose the method that looks easier for the specific problem. Substitution works well when one variable has a coefficient of one. Elimination works well when coefficients are easy to make equal. To summarize what we have learned about solving systems of equations: A system requires finding values that satisfy all equations simultaneously. The substitution method involves solving for one variable and substituting into the other equation. The elimination method makes coefficients equal and eliminates variables through addition or subtraction. Always verify your solution by checking both original equations. Choose the method that appears easier for each specific problem you encounter.