Problem 38 - Algebra 1

Question

Use substitution to I solve the linear system. Then use a graphing calculator or a computer to check your solution. $$\begin{aligned} &x-2 y=9\\\ &1.5 x+0.5 y=6.5 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 5\) and \(y = -2\).

Step 1: Rearrange one of the equations

Rearrange the first equation so that it is in terms of x, this will help us isolate the variable we want to solve for. From \(x - 2y = 9\), we can rearrange to \(x = 2y + 9\).

Step 2: Substitute the rearranged equation into the other equation

Substitute \(x = 2y + 9\) from the first equation into the second equation. This gives us \(1.5(2y + 9) + 0.5y = 6.5\). Now this equation can be solved for y.

Step 3: Solve for y

Solving for y, simplifies the equation to \(3y + 13.5 + 0.5y = 6.5\), combine like terms to get \(3.5y + 13.5 = 6.5\). Subtract 13.5 from both sides of the equation gives us \(3.5y = -7\), and dividing by 3.5 gives us \(y = -2\).

Step 4: Substitute y = -2 back in to the first equation

Substitute \(y = -2\) into the first equation \(x=2(-2) + 9\) to find \(x = 5\).

Step 5: Check the solution

Plug \(x = 5\) and \(y = -2\) into both original equations to confirm they hold true. This is an essential step to verify if the obtained values are the correct solutions.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed on a coordinate plane. A linear equation is typically composed of variables (like \(x\) and \(y\)), constants, and coefficients, all arranged in such a way that will result in a first-degree polynomial. The general form of a linear equation in two variables is expressed as \(ax + by = c\). This equation can represent a line in the geometric sense.
Linear equations are significant because:

• They can easily show relationships between variables.
• They help solve problems in various fields such as physics, economics, and engineering.
• They serve as the building block for more complex equations that one encounters in algebra.

In our exercise, we have two linear equations: \(x - 2y = 9\) and \(1.5x + 0.5y = 6.5\). Each equation represents a line, and our goal is to find where these lines intersect by solving them simultaneously.

Solving Systems

Solving systems of equations is all about finding a common solution for two or more equations. When working with linear systems, the solution represents the point where the lines intersect on a graph. There are different methods to solve such systems, including substitution, elimination, and graphing.
In the substitution method, we isolate one variable in one of the equations and substitute the expression for this variable into the other equation. This leads to a single equation with one variable, making it easier to solve. Let's break down the substitution method:

• Step 1: Rearrange one equation to express one variable in terms of the other. In our example, \(x = 2y + 9\) was obtained from rearranging the first equation.
• Step 2: Substitute this expression into the other equation to solve for the second variable. Here, \(1.5(2y + 9) + 0.5y = 6.5\) allows us to find \(y\).
• Step 3: Solve for the remaining variable and back-substitute to find the values for both variables.

This makes substitution a systematic approach to solving linear equations, especially helpful when dealing with systems of two equations.

Graphing Calculator

A graphing calculator is a valuable tool that can plot equations on an electronic screen, offering a visual way to understand and solve mathematical problems like systems of linear equations. These calculators allow for direct input of equations and can compute intersections automatically, providing immediate visual feedback.
Using a graphing calculator to check solutions for systems of equations, like the one in our exercise, involves inputting both equations into the device. By doing this, the calculator will graph both lines:

• The intersection point confirms the solution to the system.
• This serves as a quick and effective way to verify calculated answers from algebraic methods.

For the provided exercise, once you've solved the system algebraically and found \(x = 5\) and \(y = -2\), inputting these equations into a graphing calculator enables you to quickly check that these lines indeed intersect at this exact point, ensuring the reliability of your solution. Utilizing such technology can enhance understanding and accuracy in solving mathematical problems.