Problem 10 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The system has infinitely many solutions because the two equations are representing the same line.

Step 1: Checking Consistency

Divide the second equation by \(-3\) to simplify the system.\[-12x+3y=-24 → 4x - y = 8\]. Now it can be seen that it's identical to the first equation \(-4x + y = -8\).

Step 2: Evaluating the system

Having both equations identical means it's the same line in the coordinate plane. Therefore, every point on this line is a solution to the system.

Step 3: Counting the Solutions

As the two equations represent the same line, it means this linear system has infinitely many solutions.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations where you solve one of the equations for one variable and then substitute that expression into the other equation. This method is particularly useful when one equation is already solved or easily solvable for one variable.

In the given problem, we can start by rewriting the first equation in terms of one variable:

• Take the equation \(-4x + y = -8\)
• Solve for \y\, getting \y = 4x - 8\

Then, substitute \y = 4x - 8\ into the second equation. However, notice here, once simplified as done in the solution, both equations are identical. This observation means substituting one into the other doesn’t change the equation—telling us they are dependent on each other and essentially the same line when plotted.

The substitution method remains useful, as it initially detects whether different or redundant representations of lines (equations) exist in a system of equations.

Linear Combinations

Linear combinations are another approach to solve systems of equations, also known as the addition or elimination method. This process involves adding or subtracting multiple versions of the equations to eliminate a variable, making it easier to solve the system.

In our example, we start by manipulating the given equations:

• The first equation is \-4x + y = -8\.
• For the second equation, simplify \(-12x + 3y = -24\) by dividing every term by \-3\, resulting in \4x - y = 8\.

Notice that \4x - y = 8\ is the same as the first equation when the signs are flipped, meaning a proper addition or subtraction isn’t necessary because the objective of eliminating a variable is achieved by recognizing they are equivalent.

The insights acquired from using linear combinations can help determine if systems are consistent, like in this problem, leading further to the revelation of infinite solutions if the systems represent the same line.

Infinite Solutions

A system of equations with infinite solutions means that there are countless solutions that satisfy both equations. This situation arises when the equations describe the same line.

In our scenario, we noticed that both equations simplified to the same expression, indicating that they map to the same graphical line.

• Graphically, the lines overlap completely.
• Every point on the line is a solution to both equations.

This leads to the solution being infinite. Since both original equations, upon simplification or comparison, provided identical terms, it’s clear they are different visualizations of the same condition or line in a 2D coordinate system. Understanding infinite solutions help clarify mathematical consistency and dependency within systems of equations.