Problem 8 - Algebra 1: Concepts and Skills

Question

Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$\begin{array}{r} {-4 x+y=-8} \\ {-12 x+3 y=-24} \end{array}$$

Step-by-Step Solution

Answer

The system of linear equations has infinitely many solutions.

Step 1: Simplify the equations

The given system of equations is \[\begin{array}{r}{-4 x+y=-8} \{-12 x+3 y=-24}\end{array}\]We notice that we can simplify the second equation by dividing every term by -3, yielding the system: \[\begin{array}{r}{-4x + y = -8} \{4x - y = 8}\end{array}\]

Step 2: Add the simplified equations

We can add the two equations, which eliminates \( y \) from the system: \[\begin{array}{r}{-4x + y} \{+ 4x - y}\end{array}\]to get \( 0=0 \)

Step 3: Analyze the result

When the two equations are added and they yield a universal truth (\(0 = 0\)), it means that the system is dependent, and there are an infinite number of solutions. In essence, both equations represent the same line.

Key Concepts

Substitution Method

The substitution method is a popular technique used to solve linear systems. It's effective when one of the equations in the system is easily solvable for one variable. Here's how it works:

• Solve one of the equations for one variable in terms of the other.
• Substitute this expression into the other equation to find the value of a single variable.
• Use this value to find the other variable by substituting back into one of the original equations.

This method is very useful when the coefficients of variables allow for quick isolation. Its strength lies in its straightforwardness, especially when the numbers involved are simple. However, it becomes cumbersome with more complicated coefficients and when fractions are involved.

Linear Combinations

Linear combinations involve adding or subtracting equations to eliminate one of the variables, making it simpler to solve for the remaining variable. This method is also known as the elimination method. To use linear combinations:

• Multiply one or both equations by a constant so that adding or subtracting the equations will eliminate one variable.
• Solve for the remaining variable.
• Substitute back to find the other variable.

In our problem, simplifying and adding the equations to get a result like \(0 = 0\) suggests that the system might be dependent. This realisation provides a straightforward way to identify some special cases in linear systems.

Dependent System

A dependent system is when two or more linear equations represent the same line or equation graphically. In a dependent system:

• All solutions that satisfy one equation will satisfy the other.
• The graphs of the equations will overlap completely, as they are the same line.

In the given problem, after simplifying, both equations were equivalent, leading us to understand that they are essentially the same line. This is indicative of all solutions lying on this line, reflecting infinite solutions as there are countless points that lie along a line.

Infinite Solutions

Infinite solutions occur when every point on one line is also on the other, meaning that the two lines are exactly the same. When you see a statement like \(0 = 0\) after simplifying and potentially solving the equations, it indicates infinite solutions. It means that no matter what value you pick for one variable, there will be a corresponding value of the other variable that makes both equations true. This concept of infinite solutions points to the fact that there isn't a single intersection point between the equations; instead, they overlap entirely, representing countless solutions along a single line.