Problem 41 - Algebra for College Students

Question

How do we know that the graphs of \(2 x-3 y=6\) and \(-2 x+3 y=-6\) are the same line?

Step-by-Step Solution

Answer

The graphs are the same line because both equations reduce to \(y = \frac{2}{3}x - 2\).

Step 1: Understand the Equations

We have two equations: the first is \(2x - 3y = 6\) and the second is \(-2x + 3y = -6\). These equations represent lines on the Cartesian plane.

Step 2: Arrange the Equations

Our goal is to see if these two equations describe the same line. Let's rewrite both equations in slope-intercept form \(y = mx + b\) for comparison.

Step 3: Convert the First Equation

Starting with the first equation: \(2x - 3y = 6\). Subtract \(2x\) from both sides to get \(-3y = -2x + 6\). Next, divide every term by \(-3\): \[y = \frac{2}{3}x - 2\].

Step 4: Convert the Second Equation

Now with the second equation: \(-2x + 3y = -6\). Add \(2x\) to both sides yielding \(3y = 2x - 6\). Divide all terms by \(3\): \[y =\frac{2}{3}x - 2\].

Step 5: Compare the Equations

Both equations after conversion to slope-intercept form are \(y = \frac{2}{3}x - 2\). This shows that both lines have the same slope and y-intercept, indicating they are indeed the same line.

Key Concepts

Slope-Intercept Form

When analyzing or comparing lines, the slope-intercept form is a powerful tool. It helps us understand a line's steepness and where it crosses the y-axis. The slope-intercept form of a linear equation is given by:\[y = mx + b\]Where:

• \(m\) represents the slope of the line. Slope indicates the direction and steepness of the line.
• \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

For instance, when converting the equation \(2x - 3y = 6\) to slope-intercept form, we perform algebraic operations to isolate \(y\). This gives us \(y = \frac{2}{3}x - 2\), indicating a slope of \(\frac{2}{3}\) and a y-intercept of \(-2\). Achieving this form allows analysts to quickly compare lines, understand their behavior, and graph them.

Graphing Linear Equations

Graphing linear equations offers a visual representation of mathematical relationships. It transforms algebraic expressions into lines on a coordinate plane. To graph a line from its equation, knowing the slope-intercept form is beneficial. Here's how you can graph an equation like \(y = \frac{2}{3}x - 2\):

• Start by marking the y-intercept, which in this example is \(-2\). Locate this on the y-axis.
• Use the slope \(\frac{2}{3}\) to determine the next point. Slope is typically seen as a fraction \(\frac{rise}{run}\). Move up 2 units and right 3 units from the y-intercept.
• Draw a straight line through these points. Since it's linear, any further points will also form a line with consistent steepness.

Visualizing through graphs aids in comprehending the relationships between variables, making complex algebraic equations more understandable at a glance.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, melds algebraic equations and geometric concepts using a coordinate plane. It allows one to investigate the properties of geometric figures using algebraic equations.In coordinate geometry, each point on the plane is described by an \((x, y)\) coordinate. For lines, key characteristics such as slopes and intercepts are vital. In comparing the equations \(2x - 3y = 6\) and \(-2x + 3y = -6\), this branch of math confirms that both lines share the same slope \(\frac{2}{3}\) and y-intercept \(-2\). Therefore, they coincide.Coordinate geometry is not just about equations; it's about understanding relationships within the space. By applying simultaneous equations or transformations, one can assess parallelism, perpendicularity and even solve for intersection points effectively.