Problem 2 - Elementary Algebra Within Reach

Question

In Exercises 1-4, solve the system by the method of substitution. $$ \left\\{\begin{array}{l} y=-2 x+9 \\ y=3 x-1 \end{array}\right. $$

Step-by-Step Solution

Answer

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

Step 1: Isolate a variable in one of the equations

In the given set of equations, each equation has already isolated y, making it easy to set them equal to each other: \(-2x + 9 = 3x - 1\)

Step 2: Solve for x

Combine like terms and solve for x: \(-2x - 3x = -1 - 9, -5x = -10, x = -10/-5, so x = 2\)

Step 3: Substitute back into one of the original equations

Substitute x = 2 back into the first equation: \(y = -2*2 + 9 = -4 + 9\) which gives \(y = 5\)

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. The goal when working with systems is to find the values for the variables that satisfy all equations in the system simultaneously.
For example, the system provided in the exercise includes two equations with the variables \(x\) and \(y\):

• \(y = -2x + 9\)
• \(y = 3x - 1\)

To solve this system, you need to find a pair \((x, y)\) that makes both equations true at the same time. Methods such as substitution or elimination are commonly used to achieve this goal. In this exercise, the substitution method is employed, which is particularly useful when one of the variables is already isolated. This method simplifies the process by converting the system into a single equation that can be easily solved.

Solving Algebraic Equations

Solving algebraic equations involves finding the numerical values of the variables that make the equation true. For linear equations, this usually means finding where two lines intersect, as each equation can be represented as a line on a graph.
In the substitution method, we take advantage of the fact that each equation gives us the same value for \(y\). By setting the expressions for \(y\) equal, we created a single equation:

• \(-2x + 9 = 3x - 1\)

The next step is to solve for \(x\). We combine like terms and simplify:

• \(-2x - 3x = -1 - 9 \rightarrow -5x = -10\)
• Dividing each side by \(-5\) leads to \(x = 2\).

Once you find \(x\), you substitute it back into one of the original equations to find \(y\). This ensures that the solution works for the entire system.

Linear Equations

Linear equations are equations of the first degree, which means they graph as straight lines. They can take the form \(y = mx + b\), where \(m\) represents the slope, and \(b\) is the y-intercept.
Each equation in our system is a linear equation. When we talk about solving a system of linear equations, we're looking for the point where these two lines intersect on a coordinate plane.
For the given exercise, substituting \(x = 2\) into \(y = -2x + 9\), we calculate:

• \(y = -2 \times 2 + 9\)
• \(y = -4 + 9\)
• \(y = 5\)

This means the lines, represented by the equations, intersect at the point \((2, 5)\). This intersection point is the solution to the system of equations, verifying that the equations have consistent solutions and can indeed "meet" or "intersect" at one particular point.