Problem 21 - Elementary Algebra Within Reach

Question

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 0.1 x-0.1 y=0 \\ 0.8 x+0.3 y=1.5 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of equations is x = 1.36, y = 1.36

Step 1: Multiply the equations to make coefficients same

To eliminate one variable, the coefficients in front of x or y in both equations have to be the same. Multiply the first equation by 8 and second equation by 1 to get: \[ 0.8 x - 0.8 y = 0 \] \[ 0.8 x + 0.3 y = 1.5 \]

Step 2: Subtract second equation from the first

Subtract the second equation from the first to get: \[ -1.1y = -1.5 \] This simplifies to \[ y = 1.36 \]

Step 3: Substitute y=1.36 into first equation

Substitute y=1.36 into the first equation of system:\[ 0.1 x - 0.1 * 1.36 = 0 \] This simplifies to \[ x = 1.36 \]

Key Concepts

Algebraic Method

The algebraic method is a fundamental technique in solving systems of linear equations. It involves manipulating the equations using algebraic operations such as addition, subtraction, multiplication, and division. The goal is to isolate variables and find their values, thus, solving the system. In our exercise, algebraic operations are used to modify the coefficients so that one of the variables can be eliminated. This is a strategic approach, as it simplifies the problem into solvable steps. By ensuring the coefficients of one variable in both equations are the same, we set the stage to eliminate that variable.

Notably, students should remember to check the solutions in both original equations to verify their correctness, since an incorrect manipulation can lead to an incorrect solution. It's also worth noting that there are various methods within the algebraic process itself, and the elimination method is just one among them.

Variable Elimination

Variable elimination in solving systems of equations is all about strategically removing one variable, so we can solve for the other. It starts by making the coefficients of either the x or y variable identical in both equations, enabling us to subtract or add the equations together and cancel out one of the variables.

For instance, in our exercise, the coefficients of x become equal after multiplying the first equation, allowing us to subtract one equation from the other to eliminate x. Once a variable is eliminated, we find the value of the remaining variable. Knowing this value, we can substitute it back into one of the original equations to find the value of the variable that was eliminated.

Linear Equations

Linear equations form the foundation of the system under discussion. They are algebraic equations in which each term is either a constant or the product of a constant and a single variable. Linear equations can be recognized by their straight-line graph patterns on a coordinate plane. The system given in our exercise consists of two linear equations with two variables, x and y.

In the context of solving systems, the interactions of these lines—whether they intersect, are parallel, or overlap—give us vital information. Intersecting lines correspond to a single solution for the system, parallel lines indicate no solution as the lines never meet, and overlapping lines reveal an infinite number of solutions. Our system has a unique solution where the two lines, representing each equation, intersect at a single point on the graph.