Question

Solve each system of equations using any method you wish. \(\left\\{\begin{aligned} 2 x+8 y &=-8 \\ x+7 y &=-13 \end{aligned}\right.\)

Short answer

The solution is \((8, -3)\).

Step-by-step solution

Write the system of equations

The given system of equations is: Equation 1: \(2x + 8y = -8\) Equation 2: \(x + 7y = -13\)

Solve Equation 2 for x

From Equation 2: \(x + 7y = -13\), solve for \(x\). \[ x = -13 - 7y \]

Substitute \(x\) into Equation 1

Substitute \(x = -13 - 7y\) into Equation 1: \[2(-13 - 7y) + 8y = -8\]

Simplify the equation

Simplify the resulting equation: \[2(-13 - 7y) + 8y = -8\] \[-26 - 14y + 8y = -8\] Combine like terms: \[-26 - 6y = -8\]

Solve for y

Solve for \(y\): \[-26 - 6y = -8\] Add 26 to both sides: \[-6y = 18\] Divide by -6: \[y = -3\]

Substitute \(y\) back into Equation 2

Substitute \(y = -3\) back into the equation for \(x\) found in Step 2: \[x = -13 - 7(-3)\] \[x = -13 + 21\] \[x = 8\]

Write the solution

The solution to the system of equations is \((x, y) = (8, -3)\).

Key concepts

solving systems of equations

Solving systems of equations involves finding the values of variables that satisfy all equations in the system simultaneously. In this exercise, we are given a system of two linear equations and we need to solve for both variables, \( x \) and \( y \). The system of equations looks like this:
\[ 2x + 8y = -8 \] \[ x + 7y = -13 \]
These equations represent two lines in a plane. The solution to the system is the point where these two lines intersect. There are different methods to solve such systems, including graphing, substitution, and elimination.
For this problem, we will use the substitution method, which involves solving one of the equations for one variable and then substituting that expression into the other equation.

substitution method

The substitution method is a straightforward way to solve a system of equations. Here’s how we can use it:

• Solve one of the equations for one of the variables.
• Substitute this solution into the other equation.
• Simplify and solve for the remaining variable.
• Substitute back to find the first variable.

Let’s apply this to our given system:
First, solve the second equation for \( x \):
\[ x + 7y = -13 \] \[ x = -13 - 7y \]
Next, substitute this expression for \( x \) in the first equation:
\[ 2(-13 - 7y) + 8y = -8 \]
Now, simplify:
\[ -26 - 14y + 8y = -8 \] \[ -26 - 6y = -8 \]
Solve for \( y \):
\[ -6y = 18 \] \[ y = -3 \]
Finally, substitute \( y = -3 \) back into the expression for \( x \):
\[ x = -13 - 7(-3) \] \[ x = -13 + 21 \] \[ x = 8 \]
The solution to the system is \( (x, y) = (8, -3) \). This means the lines intersect at this point.

linear equations

Linear equations are equations where the highest power of the variable is one. These equations are graphed as straight lines. Examples include:
\[ 2x + 8y = -8 \] \[ x + 7y = -13 \]
In these equations, \( x \) and \( y \) are variables, and the coefficients (numbers multiplied by the variables) determine the slope of the line.
A linear equation can have:

• One solution (the lines intersect at one point).
• No solution (the lines are parallel).
• Infinitely many solutions (the lines are the same).

In our problem, the two linear equations intersect at exactly one point, \( (8, -3) \), which means there is one unique solution to the system of equations.
To summarize, understanding linear equations and how to solve them using methods like substitution can make solving more complex systems of equations much easier.