Question

Use substitution to I solve the linear system. Then use a graphing calculator or a computer to check your solution. $$\begin{aligned} &x+y=20\\\ &\frac{1}{5} x+\frac{1}{2} y=8 \end{aligned}$$

Short answer

The solution to the system of equations is \(x = 8\) and \(y = 12\).

Step-by-step solution

Isolate a variable in one of the equations

Choose one of the equations to isolate a variable. In this example, the first equation \(x+y=20\) can be used. Isolating the variable \(x\) gives us \(x=20-y\)

Substitute this expression in the other equation

Now that you have \(x\) in terms of \(y\), substitute the expression into the other equation \(\frac{1}{5}x + \frac{1}{2}y=8\). This gives us: \(\frac{1}{5}(20-y) + \frac{1}{2}y=8\)

Solve for \(y\)

Now solve the equation for \(y\). First simplify the equation: \(\frac{20-y}{5} + \frac{y}{2}=8\). After simplifying and rearranging, the equation transforms into: \(4 + \frac{1}{2}y - \frac{1}{5}y = 8\). Combining like terms and solving for \(y\) gives \(y = 12\)

Substitute \(y\) into the first equation to solve for \(x\)

Now that we have the value of \(y\), substitute it into the first equation: \(x = 20 - y\). Therefore when \(y = 12\), then \(x = 20 - 12 = 8\)

Check solution using a graphing calculator or computer replacement

Finally, substitute the values for \(x\) and \(y\) into both equations to verify they're the correct solutions. A graphing calculator or computer software will be used to graph the equations and this should show that the point of intersection is indeed \((8, 12)\), thus confirming the solution is correct.

Key concepts

Substitution Method Algebra

The substitution method is a critical technique for solving systems of linear equations, where one equation is manipulated to express one variable in terms of the other, which can then be substituted into the second equation.

To understand this method, let's consider a simple system of equations:

• Equation 1: \( x + y = 20 \)
• Equation 2: \( \frac{1}{5}x + \frac{1}{2}y = 8 \)

First, isolate one of the variables in one of the equations. For example: \( x = 20 - y \). This expression for \( x \) is then substituted into the other equation, allowing us to solve for \( y \). Once \( y \) is found, we substitute it back into our original expression to find \( x \).

Breaking Down the Equations

By breaking down the system step by step, we avoid confusion and make sure that each part of the process is understood before moving on to the next step. This method is powerful because it transforms a system of equations into a single equation in one variable, which is often simpler to solve.

Graphing Calculator Check

After solving the equations, it's essential to validate our solutions. A graphing calculator offers a practical way to check the calculations. By inputting both equations into the graphing calculator, we can visualize their graphs and confirm our solution.

With the equations from our example:

• \( x + y = 20 \)
• \( \frac{1}{5}x + \frac{1}{2}y = 8 \)

We should see that the graphs intersect at the point \( (8, 12) \), thus validating our solutions. This visual check is a powerful tool that aligns with the algebraic solution and provides additional understanding of how linear equations behave graphically.

Linear Equations Solutions

Solutions to linear equations are the values that satisfy all equations in a system simultaneously. For two-variable linear systems, the solution can be visually represented as the intersection point of two lines on a graph.

In our worked example, the substitution method led us to the solution \( (8, 12) \), meaning that when \( x = 8 \) and \( y = 12 \), both original equations are satisfied.

• In Equation 1: \( 8 + 12 = 20 \) holds true.
• In Equation 2: \( \frac{1}{5}(8) + \frac{1}{2}(12) = 8 \) also holds true.

These two values are the only pair that will satisfy both equations, making them the unique solution for this system of linear equations.

Systems of Equations

A system of equations consists of two or more equations with a common set of variables. The solution to the system is the set of variable values that satisfy all equations within the system.

In the context of our substitution example, we worked with a system of two equations and two variables: \( x \) and \( y \). Our goal was to find the specific values of \( x \) and \( y \) where the lines represented by these equations would intersect, which represents their collective solution.

Consistency and Types of Solutions

It's notable that systems of equations can be consistent (one unique solution), inconsistent (no solution, lines are parallel), or dependent (infinite solutions, lines coincide). The substitution method is just one of several strategies to find solutions, and understanding when to use it is part of grasping the broader topic of solving systems of equations.