91Ó°ÊÓ

When tackling the topic of systems of equations, it's essential to understand we're dealing with a set of two or more equations. Each of these equations contains variables, typically represented by letters such as \(x\) and \(y\). The main task is to discover the values of these variables that satisfy all equations in the system. In our exercise, we have the system \(y = -x + 6\) and \(y = x - 2\).

A system of equations can have:

• A single unique solution if the lines intersect at one point.
• No solution if the lines are parallel and never meet.
• Infinitely many solutions if the lines overlap completely, indicating they are the same line.

To solve these systems by graphing, we plot each equation on the same graph. The intersection of these lines, if any, suggests the solution or solutions to the system.

Graphing equations is a visual technique to find a system's solution. Start by plotting each line based on its equation. For \(y = -x + 6\), plot the y-intercept (0, 6), then move down 1 unit and right 1 unit to follow the slope of -1. Draw the line. For \(y = x - 2\), locate the y-intercept (0, -2), and then move up 1 unit and right 1 unit, as the slope is 1. Connect these points with a line.

The point at which these two lines meet is called the point of intersection. It's key because it represents a solution shared by both equations. In our exercise, the lines intersect at the coordinate \((4, 2)\). This means \(x = 4\) and \(y = 2\) is a solution for both equations.

Having a visual graph is an excellent way to understand how equations interact with each other. This visual representation allows us to see relationships directly, making it easier to identify solutions quickly.

Solving a system of linear equations doesn't end with identifying the intersection point. Good practice involves verifying that this point indeed solves each equation in the system. Start by plugging the intersection coordinates back into the original equations.

For our equations, substitute \((4, 2)\) back. In \(y = -x + 6\), insert \(x = 4\):

• \(y = -4 + 6 = 2\) - the equation holds.

Similarly, in \(y = x - 2\), insert \(x = 4\):

• \(y = 4 - 2 = 2\) - this equation also holds true.

By confirming both equations are satisfied, we verify that \((4, 2)\) is indeed a true solution.

Verification ensures accuracy and boosts confidence in graphically solving systems of equations. It reinforces the reliability of the graphical approach and builds trust in the solution process.