Problem 70 - Algebra 1

Question

Solve the system of linear equations if possible. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&4 y=-5 x-3\\\&15 x+12 y=9\end{aligned} $$

Step-by-Step Solution

Answer

This system of equations has no solution since the two lines represented by them are parallel and never intersect.

Step 1: Rewrite the equations in standard form

Rearrange the equations so they are in the standard form. The first equation \(4y = -5x - 3\) should be rewritten as \(5x + 4y = -3\). The second equation doesn't need any adjustment -- it's already in standard form \(15x + 12y = 9\).

Step 2: Check if the equations are multiples of each other

If one equation is a multiple of the other, the system will have either no solution or infinitely many solutions. In this case, \(5x + 4y = -3\) is not a multiple of \(15x + 12y = 9\), so the system does not have infinitely many solutions or no solutions, so it has exactly one solution.

Step 3: Solve the system of equations

To do this, one could use substitution or elimination, but in this case, the coefficients of x and y in the two equations form an easy target for the elimination method. If we multiply the first equation by 3 and the second one by 1, the two equation become: \(15x + 12y = -9\) and \(15x + 12y = 9\). Subtract the first equation from the second one, we are left with \(0 = 18\), which is not a true statement.

Step 4: Conclusion

Since the result of the subtraction leads to a false statement, it indicates that these two lines are parallel and never intersect. Therefore, this system of equations has no solution.

Key Concepts

Linear Equations

Understanding linear equations is foundational for solving systems of equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are used to represent straight lines on a coordinate graph.

These equations can typically be written in the form of ( ax + by = c ), where ( a ), ( b ), and ( c ) are constants, and ( x ) and ( y ) represent variables. Understanding how to manipulate these equations is crucial when trying to find their point of intersection, if it exists, which leads us to solving the system of equations.

Elimination Method

The elimination method is an effective algebraic technique used to find the solution to a system of linear equations. This approach focuses on eliminating one variable to solve for the other. To apply this method, you often multiply one or both of the equations by a specific number so that adding or subtracting the equations will cancel out one of the variables.

For example, if you have two equations, ( 2x + 3y = 5 ) and ( 4x - 3y = 11 ), by adding them directly, you eliminate ( y ) and can solve for ( x ). After finding the value of ( x ), you can substitute it back into one of the original equations to find the value of ( y ). This method is particularly useful when you have a system with exactly one solution.

No Solution

A system of linear equations may sometimes have no solution. This scenario arises when the lines represented by the equations are parallel – they never intersect. Algebraically, this is evidenced when attempting to solve the system results in an equation with an untrue statement, such as (0 = 18).

In the context of the exercise, we face such an event as the process of elimination led to an impossible equation, reinforcing that these two particular lines do not cross at any point. Recognizing this outcome is integral when assessing the solution set of a system of equations.

Parallel Lines

In geometry, parallel lines are lines in a plane that never meet; they remain the same distance apart over their entire length. Translating this concept into algebra, two linear equations are said to be parallel if they have identical slopes but different y-intercepts.

To determine if lines are parallel, we can compare their standard form equations. If the ratios of the coefficients of ( x ) and ( y ) (the slopes) are equal, but the constants (y-intercepts) are not, the lines are parallel. For example, equations ( ax + by = c1 ) and ( ax + by = c2 ), where ( c1 ) does not equal ( c2 ), represent parallel lines.

Algebraic Methods

Algebraic methods encompass a range of techniques used to solve equations, including systems of linear equations. In addition to the elimination method, other approaches include substitution and graphing. Each method has its strengths and is chosen based on the characteristics of the system being solved.

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. The graphing method, on the other hand, involves drawing the lines represented by the equations on a coordinate plane to find their intersection visually. These algebraic and graphical approaches form the core toolkit for students tackling linear systems.