Question

Use substitution to I solve the linear system. Then use a graphing calculator or a computer to check your solution. $$\begin{aligned} &0.50 x+0.25 y=2.00\\\ &x+y=1 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 5\) and \(y = -4\).

Step-by-step solution

Step 1:Isolating One Variable in One Equation

We will begin by expressing one variable in terms of the other. The best choice seems to be the second equation, \(x+y = 1\). We express \(x\) by rearranging the equation to get: \(x=1-y\)

Step 2:Substitute variable \(x\) in the first equation

Next, we substitute \(x\), as derived from the second equation, into the first equation. The first equation, \(0.50x + 0.25y = 2.00\), now becomes \(0.50(1-y) + 0.25y = 2.00\)

Step 3:Solving for \(y\)

We now solve this resulting equation to find the value of \(y\). Simplify the equation to get \(0.50 - 0.50y +0.25y = 2.00\), then it becomes \(-0.25y = 2.00 - 0.50\), hence \(y = -4\) after simplifying.

Step 4:Calculate \(x\)

We substitute \(y = -4\) into the second equation to find the value of \(x\). Hence \(x = 1-y\) now becomes \(x=1- (-4)\) and this results in \(x=5\)

Step 5:Graphical Verification

Finally, the solutions can be verified by graphing the equations on a calculator or computer tool and observing the intersection point. The intersection of these two lines should be at the point (5,-4). This confirms that the solution for the equation system is correct.

Key concepts

Linear Equations

Linear equations are mathematical statements that describe a straight line when plotted on a graph. They often take the form of ax + by = c, where x and y are variables and a, b, and c are constants.

In our exercise, we deal with two linear equations: 0.50x + 0.25y = 2.00 and x + y = 1. These equations represent two lines, and the solution to the system is the point where they intersect. Understanding linear equations is crucial as they are used to model real-world situations in many fields such as physics, economics, and engineering.

To solve linear systems, it's essential to find values for x and y that satisfy both equations simultaneously. This often involves manipulation such as rearranging terms and substituting one variable into the other equation.

Substitution Method

The substitution method is a technique used to solve systems of linear equations by isolating one variable in one equation and then substituting that expression into the other equation.

For example, with the system given in our exercise, we start by rearranging the second equation, x + y = 1, to isolate x: x = 1 - y. This expression for x is then substituted into the first equation in place of x. By substituting, we effectively reduce the system of two variables down to a single variable equation, which is simpler to resolve.

Once we've substituted and solved for one variable, we back-substitute that value into one of the original equations to find the value of the second variable. This step-by-step approach is systematic and prevents errors that can occur when trying to solve for both variables at once.

Graphical Verification

Graphical verification is a useful method to confirm the solution of a system of linear equations visually. By plotting each linear equation on the same coordinate axis, the solution can be identified as the point where the lines intersect.

In the context of our exercise, after calculating the values of x and y, we can graph the equations using a graphing calculator or computer software. If we've solved the equations correctly, the graphical representation will show that the two lines intersect at the point (5, -4), which corresponds to the solution we found using the substitution method.

This visual confirmation is not only satisfying, but also an excellent way to check the accuracy of our algebraic solution. It's a good practice to use whenever possible, as it provides a concrete understanding of the abstract algebra involved in solving systems of linear equations.