Problem 15 - Algebra 1

Question

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &-3 x=36\\\ &-6 x+y=1 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system is x = -12 and y = -71.

Step 1: Solve First Equation for x

To solve the first equation for x, we divide both sides of -3x = 36 by -3. This gives us x = -12.

Step 2: Substitute x in Second Equation

Next, substitute x = -12 into the second equation -6x + y = 1. This gives -6(-12) + y = 1, which simplifies to 72 + y = 1.

Step 3: Solve for y

Then, to find the value of y, subtract 72 from both sides of the equation 72 + y = 1. This gives y = 1 - 72, and hence y = -71.

Key Concepts

Substitution Method

The substitution method is a powerful way to solve systems of linear equations. It involves solving one of the equations for one variable and substituting this expression into the other equation. This technique allows you to focus on a single variable at a time, simplifying the process dramatically.

Let's look at how this method applies in practice. We started with the equations:

• \(-3x = 36\)
• \(-6x + y = 1\)

The first step was solving the first equation for \(x\). By dividing both sides by \(-3\), we determined that \(x = -12\).

Next, we replaced \(x\) with \(-12\) in the second equation. This step simplified the equation to one involving only the variable \(y\), making it much easier to solve. The substitution method is especially useful when the equations can easily be rearranged to solve for one variable.

Equations

In mathematics, equations are statements that assert the equality of two expressions. Solving equations involves finding the values of variables that make the statement true.

In our exercise, we were given two linear equations:

• \(-3x = 36\)
• \(-6x + y = 1\)

Linear equations are the simplest kind, forming straight lines when graphed. They can usually be written in the form \(ax + by = c\). Here, each equation corresponds to a line in the coordinate plane.

By solving these equations, we're essentially finding the point where these lines intersect. This intersection point gives us the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Variables

Variables are symbols used to represent numbers whose values aren't yet known. In algebra, they help us create equations and express relationships between different quantities.

In our problem, we dealt with two variables: \(x\) and \(y\). These variables stand for unknown numbers that we need to discover. Initial equations often have more than one variable, each representing a different component of a problem.

By solving the first equation, we identified that \(x = -12\). This value was then substituted into the second equation, allowing us to solve for the second variable, \(y\).

Understanding variables and how they interact in equations is crucial to solving and interpreting mathematical expressions accurately.