Problem 94 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} y=-x-1 \\ y=x+7 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((-4, 3)\).

Step 1: Identify the System of Equations

The given system of equations is: \[ y = -x - 1 \]\[ y = x + 7 \]

Step 2: Set the Equations Equal to Each Other

Since both equations are equal to \(y\), set \(-x - 1 = x + 7\) and solve for \(x\).

Step 3: Solve for \(x\)

Combine like terms to isolate \(x\):\[ -x - 1 = x + 7 \]Add \(x\) to both sides:\[ -1 = 2x + 7 \]Subtract 7 from both sides:\[ -8 = 2x \]Divide by 2:\[ x = -4 \]

Step 4: Substitute \(x\) Back into One of the Original Equations

Use the first equation \( y = -x - 1 \) and substitute \( x = -4 \):\[ y = -(-4) - 1 \]\[ y = 4 - 1 \]\[ y = 3 \]

Step 5: Write the Solution

The solution to the system of equations is \( x = -4 \) and \( y = 3 \). Therefore, the solution is the point \((-4, 3)\).

Key Concepts

substitution method

The substitution method is one of the ways to solve a system of linear equations. It involves solving one of the equations for a single variable and then substituting that expression into the other equation. This method is useful when one of the equations is already solved for one variable. For instance, in our problem, both equations are already solved for y.

Here's how the substitution method works step-by-step:

• Solve one equation for one of the variables.
• Substitute that expression into the second equation.
• Solve the resulting single-variable equation.
• Substitute the found value back into one of the original equations to find the other variable.
• Write down the solution as an ordered pair \((x, y)\).

This method ensures the solution satisfies both equations simultaneously.

linear equations

Linear equations are equations of the first degree, meaning they have variables raised to the power of one. They will graph as straight lines on a coordinate system. In the context of our exercise, we are working with the following linear equations:

\[ y = -x - 1 \] and \[ y = x + 7 \]

These equations represent straight lines when graphed. Solving a system of linear equations means finding the point where both lines intersect, which gives us the solution for both x and y. It is important to note that linear equations can have:

• One unique solution (the lines intersect at a single point).
• No solution (the lines are parallel and never meet).
• Infinitely many solutions (the lines are the same line).

system of equations

A system of equations consists of two or more equations with the same variables. The objective is to find values for the variables that satisfy all equations in the system simultaneously. In our problem, the system of equations is:

\[ y = -x - 1 \]
\[ y = x + 7 \]

To solve this system, we use the substitution method where we express both equations in terms of y. By setting the right-hand sides of each equation equal to each other, we create a new equation with one variable. Solving this new equation gives us the value of one variable, which we then substitute back into one of the original equations to find the value of the other variable.

The final solution provides a single pair of values (x, y) that satisfies both equations in the system. These values represent the coordinates where the lines of the equations intersect.