Problem 131 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} x+y=-8 \\ x-y=-6 \end{array}\right. $$

Step-by-Step Solution

Answer

x = -7, y = -1

Step 1: Solve one equation for one variable

Choose the first equation to solve for one variable. Let's solve for x from the equation: \( x + y = -8 \)Rearrange it to get: \( x = -8 - y \)

Step 2: Substitute the expression into the second equation

Substitute \( x = -8 - y \) from Step 1 into the second equation: \( (-8 - y) - y = -6 \)Simplify this equation to find y: \( -8 - y - y = -6 \)\( -8 - 2y = -6 \)

Step 3: Solve for y

Isolate y by adding 8 to both sides: \( -2y = -6 + 8 \)\( -2y = 2 \)Divide both sides by -2: \( y = \frac{2}{-2} \)\( y = -1 \)

Step 4: Substitute y into the expression for x

Substitute \( y = -1 \) back into the expression found in Step 1: \( x = -8 - (-1) \)\( x = -8 + 1 \)\( x = -7 \)

Step 5: State the solution

The solution to the system of equations is \( x = -7 \) and \( y = -1 \).

Key Concepts

substitution method

The substitution method is a useful technique for solving systems of linear equations. You first isolate one variable in one of the equations and then substitute that expression into the other equation.
This method systematically simplifies the system step by step until you find the solution.
When you isolate a variable, you're transforming the equation to express one variable in terms of the other.
Substitution replaces this transformed variable into the second equation, which allows you to solve for the remaining variable.
Afterwards, you can back-substitute to find the value of the isolated variable in the first step, making it a clear path to solving the system.

linear equations

Linear equations are algebraic expressions where each term is either a constant or the product of a constant and a single variable.
They can be written in the general form: \(Ax + By = C\).
In the example exercise, you have the system:

• \(x + y = -8\)
• \(x - y = -6\)

Each equation represents a line on a graph, and the solution to the system is the point where these lines intersect.
By understanding the properties of linear equations, like their slopes and intercepts, you can better interpret and solve problems involving them.

algebraic solution steps

Algebraic solution steps break down the process of solving equations into clear and manageable actions.
For the given system:

• Step 1: Solve one equation for one variable: Rearrange \(x + y = -8\) to get \(x = -8 - y\).
• Step 2: Substitute the expression into the second equation: Replace \(x\) in \(x - y = -6\) with \(-8 - y\): \((-8 - y) - y = -6\).
• Step 3: Simplify and solve for y: Combine like terms to get \(-8 - 2y = -6\), then isolate \(y\) to find \(y = -1\).
• Step 4: Substitute back to find x: Replace \(y = -1\) in the expression \(x = -8 - y\), resulting in \(x = -7\).
• Step 5: State the solution: The solution is \(x = -7\) and \(y = -1\).

By clearly defining each step, you make it easier to solve the equations rigorously and without confusion.