Problem 90 - College Algebra

Question

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

Step-by-Step Solution

Answer

A system of linear equations has infinitely many solutions if the two equations are equivalent, i.e., they represent the same line. You can determine this by using the addition or substitution method; if the resulting equation is always true, then there are infinitely many solutions. Graphically, this corresponds to the situation where the graphs of the two equations coincide.

Step 1: Understanding Addition and Substitution Methods

These are methods used to solve a system of linear equations. The addition method involves adding the two equations together to eliminate one of the variables, while the substitution method involves solving one equation for one of the variables, and then substituting this into the other equation.

Step 2: Determining Infinitely Many Solutions

A system of linear equations has infinitely many solutions if the two equations are equivalent, meaning they represent the same line. This can be determined if after using addition/substitution, the resulting equation is always true irrespective of the value of variables, such as 0 = 0.

Step 3: Relationship Between the Graphs

Graphically, if a system of linear equations has infinitely many solutions, the graphs of the equations will be the same line. This means that every point on the line is a solution, thus there are infinitely many solutions.

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a popular technique for solving systems of linear equations. This method involves combining two equations so that one of the variables is eliminated, making it easier to solve for the remaining variable.
To do this:

• Align the equations vertically by variable and constant term.
• Add or subtract the equations to cancel out one of the variables.
• Solve for the remaining variable.
• Substitute back into one of the original equations to find the value of the other variable.

This approach is especially useful when the coefficients of one of the variables are already opposites or can easily be made opposites by multiplying one or both of the equations by a constant.

Substitution Method

The substitution method is another effective way to solve a system of linear equations. This method involves isolating one variable in one of the equations and then substituting that expression into the other equation.
Here's how it works:

• Select one of the equations and solve for one variable in terms of the others.
• Substitute this expression into the other equation, replacing the variable you just solved.
• Solve the new equation for the remaining variable.
• Substitute back to find the value of the first variable.

This method is particularly advantageous when one of the variables has a coefficient of 1 or -1, as it simplifies the arithmetic involved in the substitution.

Infinitely Many Solutions

A system of linear equations has infinitely many solutions when the two equations describe the same line. This means they are equivalent equations, and every point on the line is a solution.
To determine this algebraically:

• Use the addition or substitution method on the system of equations.
• If the result simplifies to an identity such as \(0 = 0\), then the system has infinitely many solutions.

Graphically, both equations would produce the same line on a graph.

Equivalent Equations

Equivalent equations are different expressions of the same relationship between variables, typically achieved by multiplying or dividing both sides of an equation by the same non-zero number. Such equations describe the same line in a graph, and therefore, have the same set of solutions.
Recognizing that two equations are equivalent is key to identifying if a system has infinitely many solutions.
In practice:

• After simplifying, both equations will have the same coefficients for their variables and constants, essentially expressing the same equation.
• This confirms they are the same line graphically and thus have all solutions in common.

Graphing Linear Equations

Graphing linear equations is a visual approach to understanding solutions to systems of equations. Each linear equation can be represented as a line on a Cartesian plane.
When graphing:

• Start by finding the intercepts or using the slope-intercept form \(y = mx + b\).
• Identify and plot points to ensure accuracy.
• Draw the line through these points. Extend it across the plane.

The solution to a system is where the lines intersect:

• If they intersect at a single point, that's the one solution.
• If they are parallel, there's no solution.
• If they coincide, they have infinite solutions, showing they are equivalent.

Graphing provides a powerful visual representation of the relationships within a system of equations.