Problem 41 - Algebra 1

Question

Use substitution to I solve the linear system. Then use a graphing calculator or a computer to check your solution. $$\begin{aligned} &1.5 x-y=40.0\\\ &0.5 x+0.5 y=10.0 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \( (x, y) = (20, -10) \)

Step 1: Express Variable Y in terms of X

From the first equation, isolate y. This is done by subtracting \(1.5x\) from both sides of the equation. This gives: \( y = 1.5x - 40 \)

Step 2: Substitute for Y in the Second Equation

Next, substitute the expression obtained from Step 1 into the second equation. This gives the equation as: \(0.5x + 0.5(1.5x - 40) = 10\)

Step 3: Solve for X

Upon simplifying the equation, it boils down to \(x = 20\)

Step 4: Solve for Y

By substituting \(x = 20\) into the equation from Step 1, it can be calculated that \(y = -10\)

Step 5: Verify Solution

By plotting the original equations on a graphing calculator, it can be observed that the solution \( (20, -10) \) is indeed a point of intersection, thus confirming that it is the solution to the given system of equations

Key Concepts

Substitution Method

The substitution method is a technique used to solve a system of linear equations. It involves solving one of the equations for a single variable and then substituting this expression into the other equation. This effectively reduces the system to just one equation with one variable, making it much simpler to solve. This approach is particularly helpful when one equation is easier to manipulate.

In our example, we began by isolating the variable \(y\) in the first equation:

• We took the equation \(1.5x - y = 40\) and rearranged it to \(y = 1.5x - 40\).
• Next, we replaced \(y\) in the second equation with this expression, resulting in a simpler linear equation that only contains \(x\).

This process allows us to solve for \(x\), and after finding \(x\), we substitute back to find \(y\). It's an intuitive method that capitalizes on simplification and logical steps to reach the solution.

Graphing Calculator

A graphing calculator is an essential tool for visually understanding systems of equations. It can display graphs of equations and solve them when algebraic methods become cumbersome or to verify solutions found algebraically.

Using a graphing calculator, we plot both equations in the system, such as:

• \(1.5x - y = 40\)
• \(0.5x + 0.5y = 10\)

By doing so, it immediately becomes clear how these equations interact visually. The graph shows the lines or curves corresponding to each equation, and importantly, where they intersect.

With our solution point \((20, -10)\), a graphing calculator aids in confirming this result by showing the intersection point, which should coincide with our calculated solution. It's a powerful way to not only check answers but also understand the behavior and relation of the equations involved.

System of Equations

A system of equations is a set of two or more equations with multiple variables. The objective in solving these systems is to find an ordered pair (or triple, quadruple, etc., depending on the variables) that satisfies all equations simultaneously.

In our example, the system consists of the two linear equations:

• \(1.5x - y = 40\)
• \(0.5x + 0.5y = 10\)

Each equation represents a line on the coordinate plane. The solution to the system, if it exists, is the point where these lines intersect—meaning both equations share a common solution at this point. Solving systems of equations is vital in a variety of applications, including economics, physics, and engineering, where multiple factors must be satisfied concurrently.

Intersection Point

The intersection point of two lines or curves on a graph is the point where both equations are true simultaneously. This point visually represents the solution to a system of equations when graphed.

For the system in question, solving algebraically, we identified the intersection point as \((20, -10)\). When graphing the equations, this point is where both lines meet.

• By setting the equations equal to each other, we determined they intersect at this solution point.
• The intersection verifies that both equations yield the same pair of values when substituted back, thus complying with the definition of a solution to a system of equations.

Understanding intersection points is critical for grasping how systems of equations are solved and verified, both algebraically and visually through graphing.