Question

Choose a method to solve the linear system. Explain your choice, and then solve the system $$ \begin{aligned} &0.2 x-0.5 y=-3.8\\\ &0.3 x+0.4 y=10.4 \end{aligned} $$.

Short answer

The solution to the system is \(x = 70.44\) and \(y = 35.776\)

Step-by-step solution

Eliminate Decimals

Start by getting rid of decimals. Multiply Equation 1 by 10 and Equation 2 by 10. This will result in:(1) \(2x - 5y = -38\)(2) \(3x + 4y = 104\)

Equate via Elimination Method

Next, we will use the elimination method. The aim is to eliminate one variable by adding or subtracting the two equations. In our case, we can eliminate \(y\) by multiplying the first equation by 3 and the second equation by 5 and then subtract the second equation from the first. This gives us:\(6x - 15y = -114\)\(15x + 20y = 520\)Now we subtract the second from the first:\(-9x = -634\)

Solve for x

Divide both sides of the equation by -9 to solve for x: \(x = 70.44\)

Solve for y

Substitute \(x = 70.44\) into the first equation \(2x - 5y = -38\) and solve for \(y\):\(2*(70.44) - 5y = -38\)\(140.88 - 5y = -38\)\(-5y = -38 - 140.88\)\(-5y = -178.88\)Divide by -5:\(y = 178.88 / 5\)which gives:\(y = 35.776\)

Check the solutions

Substitute the found values \(x = 70.44\) and \(y = 35.776\) back into the original equations to verify that they hold true. You will find that with these values, both equations' left side will be equal to the right side.

Key concepts

Elimination Method

The elimination method is a systematic approach to solving linear systems by removing one of the variables, thereby making it simpler to solve for the other. In essence, elimination involves combining two linear equations in such a way that one variable cancels out.

Let's consider an example where this method is applied. Given two equations, the steps usually involve:

• Multiplying each equation by a suitable number so that the coefficients of one variable are opposites.
• Adding or subtracting the equations to eliminate that variable.
• Solving the resulting single-variable equation.
• Using the value of the known variable to find the value of the other variable.

In our textbook problem, after multiplying the equations to achieve opposite coefficients for the variable y, we subtract them, effectively eliminating y and simplifying the system to one equation with one unknown, x.

Solving for Variables

Solving for variables is the process of finding the values of the unknowns in equations that make the equations true. The key to solving for variables is isolation, which entails manipulating the equation until the variable of interest is by itself on one side of the equation.

For instance, in the provided solution, once we eliminate one variable using the elimination method, we get a single equation in one variable. We isolate x by dividing both sides of the equation by the coefficient in front of x, thus finding its value. Similarly, to find y, we substitute the value of x into any of the original equations and isolate y through similar algebraic manipulation. This stepwise approach simplifies the process and ensures accurate solutions.

Linear Equations

Linear equations are mathematical statements that show the equality of two expressions involving constants and variables, usually with each variable raised to the power of one. Such equations form straight lines when graphed on a coordinate plane.

The general form of a linear equation in two variables, x and y, is ax + by = c, where a, b, and c are constants. The solution to a system of linear equations is the point or points where the graphs of the equations intersect.

The essence of working with linear equations in a system is to find the values of the variables that satisfy all equations simultaneously. The elimination method is one of several techniques used to find solutions to these systems, with the objective being to reduce the system to simpler single-variable equations that are more straightforward to solve.