Problem 10 - Thinking Mathematically

Question

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=x+1 \\ y=3 x-1\end{array}\right.\)

Step-by-Step Solution

Answer

The solution for the system of equations is the intersection point of the two graph lines.

Step 1: Graphing the first equation

The first equation is a linear equation \(y=x+1\), this equation can be rewritten as \(y=1*x+1\), indicating that its slope (m) is 1 and its y-intercept (b) is 1. Plot this line on a graph.

Step 2: Graphing the second equation

The second equation is another linear equation \(y=3*x-1\), this equation can be rewritten as \(y=3*x-1\), indicating that its slope (m) is 3 and its y-intercept (b) is -1. Plot this line on the same graph as the first line.

Step 3: Finding the intersection point

The intersection point of the two lines is the solution for the system of equations. By comparing the two graphs, find the point where they intersect.

Step 4: Validating the intersection point

Once the intersection point has been found, check these coordinates by substituting them into both the original equations. If the coordinates satisfy both equations, then they are the solution of the system of equations.

Key Concepts

Graphical Solution

When we solve a system of equations graphically, we represent each equation as a line on a coordinate plane. The graphical solution method gives us a visual representation of where the solutions to the equations meet. Essentially, each line represents all the solutions for that respective equation. By finding where these lines intersect, we locate the common solution that satisfies both equations simultaneously.

This method is very useful when visualizing how two equations interact, especially when dealing with linear equations. When graphing, ensure each line is accurately plotted to make identifying the intersection easier. However, remember that graphing might not be as precise as algebraic methods if the intersection point doesn't line up neatly with grid lines.

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed. A standard form of a linear equation is \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. Linear equations describe a constant rate of change between the variables, making them predictable and easy to work with.

In the original exercise, we had two linear equations: \(y = x + 1\) and \(y = 3x - 1\). Each equation describes a straight line with different slopes and y-intercepts, resulting in two distinct lines when graphed. Understanding how to form and interpret these equations is crucial to finding their intersection when solved graphically.

Intersection of Lines

The intersection of two lines is a point on the graph where the lines cross. This specific point represents the set of coordinates \(x, y\) that satisfies both linear equations at once. In the context of a system of equations, finding this intersection is equivalent to finding the solution that makes both equations true.

In the exercise, after graphing the two lines, the intersection point is observed where they cross each other. This graphical intersection provides a quick solution to the system of linear equations. Once the intersection is visually identified, you typically verify this solution by substituting the coordinates back into the original equations.

Slope and Y-Intercept

Understanding the slope and y-intercept is essential for graphing linear equations and analyzing how lines appear on the coordinate plane. The slope \(m\) of a line indicates the steepness and the direction of the line. For instance, a positive slope means the line inclines upward, and a negative slope suggests the line declines downward.

The y-intercept \(b\), on the other hand, is the point where the line crosses the y-axis. Determining these components is crucial for graphing, as they influence where and how the line is positioned. In the exercise, the first equation had a slope of 1 and a y-intercept of 1, while the second equation had a slope of 3 and a y-intercept of -1. Knowing the differences in slopes and y-intercepts helps predict how the lines will interact and eventually determine where they might intersect on the graph.