Question

The lines graphed in the \(x y\)-plane above represent a system of two linear equations. What is the solution \((x, y)\) to the system? A) \((-1,-6)\) B) \((0,-3)\) C) \((2,3)\) D) \((3,0)\)

Short answer

After analyzing the given options and testing each one to see if it satisfies both linear equations, we find that the solution to the system is option C, \((2,3)\).

Step-by-step solution

Analyze the given options

We have been given four different options, and we need to find the one that satisfies both linear equations. Let's check each option one by one.

Test option A

Let's test option A, which is \((-1,-6)\). If this point lies on both lines, it would be the solution to the system. Substitute the \(x\) and \(y\) values into the equations of both lines and see if they hold true. If not, move to the next option.

Test option B

Now, let's test option B, which is \((0,-3)\). If this point lies on both lines, it would be the solution to the system. Substitute the \(x\) and \(y\) values into the equations of both lines and see if they hold true. If not, move to the next option.

Test option C

Now, let's test option C, which is \((2,3)\). If this point lies on both lines, it would be the solution to the system. Substitute the \(x\) and \(y\) values into the equations of both lines and see if they hold true. If this satisfies both equations, we have found the solution.

Test option D (if necessary)

If none of the previous options are the solution, we will test option D, which is \((3,0)\). If this point lies on both lines, it would be the solution to the system. Substitute the \(x\) and \(y\) values into the equations of both lines and see if they hold true. If this satisfies both equations, we have found the solution. After testing all of the given options, we will find the one that satisfies both linear equations and represents the solution to the system.

Key concepts

Linear Equations

Linear equations are mathematical statements that depict a straight line when graphed on a coordinate plane. They typically have the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. These equations have two variables, usually \( x \) and \( y \).

Some important characteristics of linear equations include:

• They represent a constant rate of change.
• Their graph is always a straight line.
• They can have one, none, or infinitely many solutions when part of a system.

Linear equations are used to model and solve real-world problems like predicting expenses, calculating distances, and many more.

In our exercise, we are dealing with a system of two linear equations graphed on a coordinate plane. The challenge is to find the point where both lines intersect.

Coordinate Plane

The coordinate plane is a two-dimensional plane defined by an x-axis (horizontal) and a y-axis (vertical). These axes intersect at a point called the origin, which has the coordinates (0,0).

The coordinate plane is divided into four quadrants:

• Quadrant I: positive \( x \) and \( y \) values
• Quadrant II: negative \( x \) and positive \( y \) values
• Quadrant III: negative \( x \) and \( y \) values
• Quadrant IV: positive \( x \) and negative \( y \) values

This system is useful for graphing algebraic equations like linear equations, helping us to visualize relationships between variables. In the problem we're solving, we graph the linear equations on the coordinate plane to locate the intersection point.

Substitution Method

The substitution method is a way to solve a system of equations by solving one of the equations for one variable and then substituting that expression into the other equation.

Here's how it works:

• Solve one equation for one variable, either \( x \) or \( y \).
• Substitute this expression into the other equation.
• Solve the resulting equation for the other variable.
• Use this solution to find the value of the first variable.

This method is particularly helpful when one of the equations is easily solved for one variable. However, in our exercise, we used a more graphical approach to find the solution. The substitution method is another valid strategy that can be employed based on equation complexity and preference.

Graphing Systems of Equations

Graphing systems of equations involves plotting each equation on the same coordinate plane and identifying their points of intersection. The intersection point is the solution to the system.

To graph each equation:

• Find at least two points by plugging in values for one variable and solving for the other.
• Draw straight lines through these points for each equation.
• Look for the point where the lines intersect.

The intersection provides the values for \( x \) and \( y \) that satisfy both equations simultaneously. In this exercise, checking each point helps see if it truly lies on both lines, ensuring it is the correct solution. This method visually demonstrates how solutions behave, reinforcing the abstract process of solving equations with tangible results.