Problem 3 - Understanding Elementary Algebra with Geometry

Question

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{r} 2 x+y=5 \\ x-y=4 \end{array}\right.$$

Step-by-Step Solution

Answer

\( x = 3 \), \( y = -1 \)

Step 1: Setup the System

Rewrite the system of equations in a stacked form to easily apply the elimination method: 1) \( 2x + y = 5 \) 2) \( x - y = 4 \)

Step 2: Align and Add Equations

Add the two equations together to eliminate the variable \( y \): \( (2x + y) + (x - y) = 5 + 4 \)

Step 3: Simplify the Result

Simplify the left side and the right side of the equation: \( 3x = 9 \)

Step 4: Solve for \( x \)

Solve for \( x \) by dividing both sides of the equation by 3: \( x = \frac{9}{3} \) \( x = 3 \)

Step 5: Substitute to Solve for \( y \)

Substitute \( x = 3 \) back into the second original equation to solve for \( y \): \( 3 - y = 4 \) \( -y = 4 - 3 \) \( -y = 1 \) \( y = -1 \)

Step 6: Verify the Solution

Substitute \( x = 3 \) and \( y = -1 \) back into the first original equation to verify: \( 2(3) + (-1) = 5 \) \( 6 - 1 = 5 \) It holds true, so the solution is verified.

Key Concepts

system of equations

A system of equations is a set of two or more equations with the same variables. The purpose is to find values for these variables that make all equations true simultaneously. In this case, we have a system of two linear equations: 2x + y = 5 and x - y = 4. Here, the variables are x and y, and we need to find their values. These types of problems are common in algebra and have practical applications in various fields, such as physics, economics, and engineering.

solving linear equations

Solving a linear equation involves finding the value of the variable that makes the equation true. Linear equations are in the form ax + b = c, where 'a', 'b', and 'c' are constants. For example, in the equation 2x + y = 5, we need to isolate the variable x or y.
Easy steps to solve linear equations:

• Combine like terms if necessary.
• Use addition or subtraction to isolate the variable term on one side of the equation.
• Divide or multiply to solve for the isolated variable.

Always remember to perform the same operation on both sides of the equation to maintain equality.

algebraic methods

Algebraic methods are techniques used to manipulate equations to find the values of unknown variables. The elimination method, used in the above exercise, is one of these techniques. It involves adding or subtracting equations to eliminate one of the variables and solve for the remaining one.
Steps for using the elimination method:

• Align the equations to easily spot coefficients.
• Multiply one or both equations to make the coefficients of one variable equal (if needed).
• Add or subtract the equations to eliminate one variable.
• Solve the resulting single-variable equation.
• Substitute the found value back into one of the original equations to find the other variable.

Another common algebraic method is substitution, where one variable is isolated in one equation and substituted into the other.