Problem 1 - Understanding Elementary Algebra with Geometry

Question

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} x+y=1 \\ x-y=3 \end{array}\right.$$

Step-by-Step Solution

Answer

x = 2, y = -1

Step 1 - Write the System of Equations

The given system of equations is: \ \( x + y = 1 \) \ \( x - y = 3 \)

Step 2 - Add the Equations

Add the two equations together to eliminate one variable: \ \( (x + y) + (x - y) = 1 + 3 \) This simplifies to: \ \( 2x = 4 \)

Step 3 - Solve for x

Divide both sides of the equation by 2 to solve for \( x \): \ \( x = 2 \)

Step 4 - Substitute x Back into One of the Original Equations

Substitute \( x = 2 \) into the first equation \( x + y = 1 \): \ \( 2 + y = 1 \)

Step 5 - Solve for y

Subtract 2 from both sides to solve for \( y \): \ \( y = 1 - 2 \) \ \( y = -1 \)

Step 6 - Check the Solution

Ensure the solution \( x = 2 \) and \( y = -1 \) satisfies both original equations: \ For \( x + y = 1 \): \ \( 2 - 1 = 1 \) (True) \ For \( x - y = 3 \): \ \( 2 - (-1) = 3 \) (True)

Key Concepts

Solving Systems of Equations

A system of equations consists of two or more equations with the same set of variables. Solving these systems means finding the set of values for the variables that satisfy all the equations simultaneously.

There are multiple methods to solve systems of equations:

• Graphical Method - Plotting the equations on a graph and finding their intersection points.
• Substitution Method - Solving one equation for one variable and substituting that solution in the other equation.
• Elimination Method - Adding or subtracting equations to eliminate one of the variables, making it easier to solve for the remaining variable.

The elimination method is particularly useful when the coefficients of one of the variables are the same or opposites. In our example, we added two equations to eliminate the variable y.

Substitution

The substitution method involves solving one of the equations for one variable, and then replacing that variable in the other equation with the solved value. This method simplifies solving for the remaining variable by reducing the number of variables in the equation.

In our solved example, after we found that x = 2 using the elimination method, we substituted this value into one of the original equations to find y. Here's how it works:

• Step 1: Choose an equation and solve for one variable. (In elimination, this step might come after addition or subtraction.)
• Step 2: Substitute the solved value into the other equation.
• Step 3: Simplify and solve for the remaining variable.

This technique ensures that both equations are satisfied with the found values. After substitution, we saw that y = -1.

Linear Equations

Linear equations are equations of the first degree, which means they have no exponents higher than one. They graph as straight lines, and a system of linear equations represents multiple straight lines.

The general form of a linear equation is: \[ ax + by = c \], where a, b, and c are constants, and x and y are variables.

To solve systems of linear equations, we can use either the elimination method or substitution method. Each technique leverages the property that linear equations graphically represent straight lines that intersect at points, and those points of intersection represent the solutions to the system.

In our example, the system of equations: \( x + y = 1 \) \( x - y = 3 \), consists of two linear equations representing two straight lines. When solved correctly, as we did, x = 2 and y = -1 is the point of intersection that satisfies both equations.