Problem 130 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} x+y=-1 \\ x-y=-5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (-3, 2)\).

Step 1: Solve one equation for a variable

Start by solving the second equation for one of the variables. Let's solve for \( x \):equation 1: \( x - y = -5 \)Adding \( y \) to both sides:\[ x = y - 5 \]

Step 2: Substitute into the other equation

Next, substitute the expression for \( x \) from Step 1 into the first equation: equation 2: \( x + y = -1 \)Substitute \( x = y - 5 \):\[ (y - 5) + y = -1 \]

Step 3: Simplify and solve for the variable

Combine like terms and solve for \( y \):\[ 2y - 5 = -1 \]Add 5 to both sides:\[ 2y = 4 \]Divide by 2:\[ y = 2 \]

Step 4: Substitute back to find the other variable

Now substitute \( y = 2 \) back into the expression found in Step 1 for \( x \):\[ x = y - 5 \]\[ x = 2 - 5 \]\[ x = -3 \]

Step 5: Write the solution as an ordered pair

The solution to the system of equations is \((x, y) = (-3, 2)\).

Key Concepts

Substitution Method

The substitution method is used to solve systems of equations. A system of equations consists of two or more equations with the same set of variables. In our example, we have two linear equations with variables x and y. The steps to use the substitution method are straightforward:
First, solve one of the equations for one variable. For instance, from the equation \( x - y = -5 \), we isolate x by adding y to both sides: \( x = y - 5 \).
Next, substitute this expression for x into the other equation. In this case, substitute \( x = y - 5 \) into \( x + y = -1 \): \( (y - 5) + y = -1 \). This transforms the system into a single equation that only includes y.
Then, solve for y. Once you have the value of y, use it to find x. Substitute y back into the equation \( x = y - 5 \): \( x = 2 - 5 = -3 \).
The solution to the system will be the ordered pair \(( x, y ) = (-3, 2)\).
The substitution method simplifies solving systems by breaking everything down into simpler steps.

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed. They have the general form \( Ax + By = C \), where A, B, and C are constants. In our example, we have two linear equations: \( x + y = -1 \) and \( x - y = -5 \).
Features of linear equations:

• They graph as straight lines.
• They can represent relationships between two variables.
• The solutions to the system are the points where the lines intersect.

By solving the system, we discover the point where both lines cross. For instance, both equations intersect at the point \( (-3, 2) \).
Linear equations are fundamental in algebra and help model real-world situations with constant rates of change.

Ordered Pairs

When we solve a system of equations, the solution is often given as an ordered pair. An ordered pair represents a specific point on the coordinate plane, where the x-value comes first and the y-value second: \(( x, y )\).
In our example, the ordered pair solution is \( (-3, 2) \), meaning that \( x = -3 \) and \( y = 2 \).
Ordered pairs:

• Identify precise points in 2D space.
• Show where equations or lines intersect.
• Represent solutions to systems of linear equations.

Understanding how to read and write ordered pairs is crucial for graphing and interpreting algebraic solutions. The ordered pair not only gives a solution but also illustrates the relationship between the variables at that point.