Problem 45 - Precalculus

Question

Solve each system of linear equations by graphing. $$\begin{aligned} &y=-x\\\ &y=x \end{aligned}$$

Step-by-Step Solution

Answer

The solution is the point (0,0) where both lines intersect.

Step 1: Understand the Equations

We have two linear equations: \( y = -x \) and \( y = x \). These equations represent two lines in the Cartesian plane.

Step 2: Identify Slopes and Intercepts

The slope of the line in the equation \( y = -x \) is -1 with a y-intercept of 0. For the line \( y = x \), the slope is 1 and the y-intercept is also 0.

Step 3: Graph the First Equation

Graph the line \( y = -x \). Start at the y-intercept (0,0) and use the slope -1 to plot another point, such as (-1,1). Draw the line through these points.

Step 4: Graph the Second Equation

Graph the line \( y = x \). Start at the y-intercept (0,0) and use the slope 1 to plot another point, such as (1,1). Draw the line through these points.

Step 5: Find the Intersection

Observe the graph to identify where the two lines intersect. Both lines intersect at the origin point (0,0).

Step 6: Verify the Solution

Check that the point (0,0) satisfies both equations. Substitute 0 for \( y \) and \( x \) in both equations: \( 0 = -0 \) and \( 0 = 0 \), both are true.

Key Concepts

System of Linear Equations

A system of linear equations is essentially a collection of two or more linear equations with the same set of variables. In our case, that means both equations involve the variables \(x\) and \(y\). When faced with such a system, the goal is to find a set of values for these variables, making both equations true simultaneously.

• If both lines intersect at a point, that point is their unique solution.
• If the lines are parallel, they have no points in common, so the system has no solution.
• If the lines overlap completely (are the same line), they have infinitely many solutions.

By graphing each line, we visually observe where they cross, providing a simple way to solve the system.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation as \(y = mx + c\), where \(m\) represents the slope and \(c\) represents the y-intercept of the line. This form is quite useful when it comes to graphing linear equations since it gives direct information on how the line behaves.
Let's break it down:

• Slope (\(m\)): This tells us how slanted the line is. For every increase of 1 in \(x\), \(y\) changes by the amount of \(m\). Positive slopes rise to the right, while negative slopes fall.
• Y-Intercept (\(c\)): This is where the line crosses the y-axis. It is the value of \(y\) when \(x\) is 0.

In our example, the equations \(y = -x\) and \(y = x\) have slopes of -1 and 1, respectively, and both intersect the y-axis at (0,0). This makes them simple yet illustrative examples.

Intersection of Lines

The intersection of two lines on a graph represents their common point. For the system of linear equations \(y = -x\) and \(y = x\), the point of intersection is crucial because it is the solution to the system. This particular intersection occurs at the origin (0,0), the point where both lines cross on the Cartesian plane.
To find this point graphically:

• Plot both lines by using their respective slopes and y-intercepts.
• Look for the point where both graphed lines intersect.

The intersection point is verified by plugging back the coordinates \((x,y)\) into both equations to check if they hold true. The intersection is a powerful visual clue in understanding how equations relate to each other.