91Ó°ÊÓ

When we talk about the points of intersection, we are referring to the specific points where two graphs meet or cross each other on a coordinate plane.

• In a system with two linear equations, the point where they intersect is the solution to the system.
• This point satisfies both equations; when you substitute the coordinates of the intersection point into both equations, they will hold true.

In our exercise, the solution to the system, and thus the point of intersection, is (-2, -1). At this point, both equations, \(3x-2y=-4\) and \(4x+2y=-10\), are satisfied when you substitute \(x=-2\) and \(y=-1\). Knowing how to identify and calculate the points of intersection helps us understand how different equations relate graphically.

Linear equations are fundamental in algebra and involve expressions of the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

• They represent straight lines on a graph.
• Their general solution is typically a set of values for \(x\) and \(y\) that make the equation true.

In our exercise, we are dealing with two such equations: \(3x - 2y = -4\) and \(4x + 2y = -10\). These equations can represent lines in the coordinate plane, and finding their intersection means finding the solution set that satisfies both equations at once. Linear equations are called linear because their graphs produce lines or, more precisely, linear paths across the graph.

Solving equations involves finding all values for the variables that make the equation true.

• In this context, solving our system involves determining the values for \(x\) and \(y\) that satisfy both linear equations simultaneously.

Follow these steps:1. **Combine both equations** to eliminate one variable. In the provided solution, adding the two equations together eliminates \(y\). \(3x - 2y + 4x + 2y = -4 + (-10)\) simplifies to \(7x = -14\).2. **Solve for one variable.** Divide both sides by 7, resulting in \(x = -2\).3. **Substitute this value into one of the original equations to find the other variable.** Plugging \(x = -2\) into the first equation gives \(3(-2) - 2y = -4\), solving for \(y\) results in \(y = -1\). By following these steps, we solve the system of equations and find their point of intersection, showcasing the interplay between algebra and geometry.

Graphing is a visual way of representing equations, and it's particularly helpful in understanding how equations behave. In the context of systems of equations:

• Graphing each equation on the same set of axes allows us to see where they intersect.
• The intersection point on the graph gives a visual representation of the solution to the system of equations.

To graph the given linear equations, we plot each equation as a line. For instance, the equation \(3x - 2y = -4\) can be graphed by finding points that satisfy the equation and connecting them to form a straight line. Similarly, we do the same for \(4x + 2y = -10\).Where these two lines cross is the solution point, \((-2, -1)\), telling us where both conditions of the equations are met in graphical form. Graphing not only aids in visualizing solutions, but it also helps in understanding the relationships and interactions between different equations.