Question

If the only solution to a system of two linear equations containing two variables is \(x=3, y=-2,\) then the graphs of the lines in the system intersect at the point ______.

Short answer

(3, -2)

Step-by-step solution

Identify the Solution Point

The problem states that the only solution to the system of equations is given by the values of variables: \(x = 3\) and \(y = -2\).

Determine the Intersection Point

In a system of linear equations, the solution \( (x, y) \) represents the point where the graphs of the two equations intersect. Thus, if \(x = 3\) and \(y = -2\) are the values that satisfy both equations, the intersection point is \((3, -2)\).

State the Intersection Point

From the identified values, the intersection point of the two lines in the system is \((3, -2)\).

Key concepts

solution to linear equations

A solution to a system of linear equations is an ordered pair \((x, y)\) that makes both equations in the system true. In other words, when you substitute the values of \((x, y)\) into both equations, they should satisfy both equations simultaneously. For example, if the given solution to a system is \(x = 3\) and \(y = -2\), then these values satisfy both linear equations in the system.

It is crucial to understand that not all systems of linear equations have a single unique solution. Some systems have no solution (parallel lines that never meet), while others have infinitely many solutions (the same line). To identify the solution correctly, you need to solve both equations and check the values you get for \((x, y)\). Often, you can use methods such as substitution, elimination, or graphing to find the solution.

graphs of linear equations

Graphs of linear equations are straight lines. Each linear equation can be written in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. When graphing a linear equation, start by plotting the y-intercept on the coordinate plane. Then, use the slope to find another point and draw the line through these points.

In graphing systems of linear equations, you are typically dealing with two lines. The point where these two lines intersect represents the solution to the system. If the lines are parallel, there is no intersection, meaning the system has no solution. If the lines are coincident (one on top of the other), there are infinite solutions.

intersection point

The intersection point of two linear equations in a system is the point where both lines meet. This point is also known as the solution to the system of equations. To find this point, you can graph both equations on the same coordinate plane and identify their meeting point.

Let's say we have the system of equations with the solution \(x = 3\) and \(y = -2\). This means the graphs of these equations intersect at the point \((3, -2)\). This point is where both equations are true, and thus, both lines pass through this coordinate on the plane. Knowing how to find and interpret the intersection point is essential for understanding how systems of linear equations work and how different solutions can be visualized geometrically.