Problem 76 - Intermediate Algebra

Question

a. Without graphing, how can you tell that the graphs of \(y=2 x+1\) and \(y=3 x+2\) intersect? b. Without graphing, how can you tell that the graphs of \(y=2 x+1\) and \(y=2 x+2\) do not intersect? c. Without graphing, how can you tell that the graphs of \(y=2 x+3\) and \(2 y=4 x+6\) are the same line?

Step-by-Step Solution

Answer

a. Solve for intersection to find a point; b. Lines have same slope (parallel) with different intercepts; c. Equations simplify to same line.

Step 1: Understanding Line Intersections

For graphs of two lines to intersect, their equations must have a common solution (i.e., a point that satisfies both equations).

Step 2: Solving for Intersection of Different Slopes

Given lines: \(y = 2x + 1\) and \(y = 3x + 2\), we solve to find common solutions by setting \(2x + 1 = 3x + 2\).

Step 3: Simplifying the Equation

Rearranging gives \(2x + 1 - 2 = 3x\) or \(-1 = x\).

Step 4: Finding Intersection Point

Substitute \(x = -1\) back into \(y = 2x + 1\) to find \(y = -1\). Thus, the lines intersect at \((-1, -1)\).

Step 5: Examining Parallel Lines

For lines \(y = 2x + 1\) and \(y = 2x + 2\), compare their slopes (both are 2). Parallel lines with different y-intercepts, 1 and 2 in this case, do not meet.

Step 6: Identifying Coincident Lines

Given \(y = 2x + 3\) and \(2y = 4x + 6\), simplify the second equation by dividing all terms by 2 to get \(y = 2x + 3\), hence both equations are identical.

Key Concepts

Intersection of Lines

When we talk about the intersection of lines, we're referring to the point where two lines meet on a graph. This is called the 'point of intersection.' You can find this point by identifying a solution that satisfies both line equations. For linear equations like \(y = 2x + 1\) and \(y = 3x + 2\), the intersection is found by setting the equations equal to each other.\( 2x + 1 = 3x + 2 \). Solve for \(x\) to find \(x = -1\).
Once we have \(x\), substitute it back into one of the original equations, say \(y = 2x + 1\), which results in \(y = -1\). Therefore, these lines intersect at the point \((-1, -1)\).

• The intersection point is where both equations result in the same value.
• It requires both lines having different slopes.
• Even without graphing, setting equations equal is the key to finding intersections.

Parallel Lines

Parallel lines are lines in a plane that never meet; they have the same slope but different y-intercepts. For example, consider the equations \(y = 2x + 1\) and \(y = 2x + 2\). Both have a slope of 2, meaning they climb at the exact same rate.
The y-intercepts, however, are 1 and 2, respectively. This means that though they are parallel and lie evenly spaced on a graph, they will never cross each other.

• Parallel lines have identical slopes.
• Different y-intercepts ensure they do not meet.
• Verifying slopes is a quick way to identify parallel lines without graphing.

Identical Lines

Identical lines are essentially the same line. It means every point on one line exactly overlaps with every point on the other. Consider the examples \(y = 2x + 3\) and \(2y = 4x + 6\). By simplifying the second equation by dividing every term by 2, we get \(y = 2x + 3\), proving they are identical.
This simplification shows that all aspects of the two equations remain constant, making them one and the same on a graph.

• Identical lines have both the same slope and y-intercept.
• Simplifying equations can reveal identical lines.
• Without graphing, algebra simplification shows line identity.