Problem 69 - Algebra A Combined Function

Question

The ordered pair (-2,3) is a solution of the three linear equations below: \(x+y=1\) \(2 x-y=-7\) \(x+3 y=7\) If each equation has a distinct graph, describe the graph of all three equations on the same axes.

Step-by-Step Solution

Answer

All lines intersect at the point (-2, 3) on the graph.

Step 1: Identify Lines from Equations

Each equation represents a straight line in the Cartesian plane. We will identify them:\(x+y=1\) is Line 1, \(2x-y=-7\) is Line 2, and \(x+3y=7\) is Line 3.

Step 2: Verify Solutions

Substitute \(x = -2\) and \(y = 3\) into each equation to verify:1. For \(x+y=1\) : \(-2+3=1\), true.2. For \(2x-y=-7\) : \(2(-2)-3=-7\), true.3. For \(x+3y=7\) : \(-2+3(3)=7\), true.Thus, the point \((-2, 3)\) is a solution for all equations.

Step 3: Graph Each Equation Individually

Graph each line on the Cartesian plane: - For \(x+y=1\), rearrange to \(y=-x+1\). The line passes through the y-intercept \((0,1)\) and has a slope of \(-1\).- For \(2x-y=-7\), rearrange to \(y=2x+7\). The line passes through the y-intercept \((0,-7)\) and has a slope of \(2\).- For \(x+3y=7\), rearrange to \(y=-(1/3)x+7/3\). The line passes through the y-intercept \((0,7/3)\) with a slope of \(-1/3\).

Step 4: Describe the Graph of All Lines

All three lines are straight lines that intersect at the point \((-2, 3)\). Since this point is a solution to all three equations, the graphs of the lines all cross each other at this single, shared point. This intersection confirms that \((-2, 3)\) is their common solution.

Key Concepts

Cartesian plane

The Cartesian plane is a two-dimensional surface defined by two axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, denoted as (0,0). The plane is divided into four quadrants:

• Quadrant I: where both x and y are positive.
• Quadrant II: where x is negative and y is positive.
• Quadrant III: where both x and y are negative.
• Quadrant IV: where x is positive and y is negative.

Each point on the Cartesian plane can be represented by an ordered pair (x, y), which indicates its exact location relative to the origin. In our exercise, we're examining the solutions of linear equations within this plane. This enables us to visualize how the lines described by these equations interact with one another.

graph of equations

When plotting equations, especially linear ones like in our exercise, we visualize them as lines on the Cartesian plane. The power of graphing lies in its ability to show the relationship between variables. Each linear equation, like \(x + y = 1\), represents a line with all the solutions as points along this line. These three equations from the exercise create distinct lines on the plane.

• Each line is straight, which is a characteristic of linear equations.
• They can be extended infinitely in both directions.
• The intersection point of these lines reflects a common solution, known as their point of concurrency, here identified as (-2, 3).

By graphing these equations, it becomes clearer how they relate or intersect, offering insight into their solutions.

slope-intercept form

The slope-intercept form is a method of expressing linear equations in the form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept. This form is particularly helpful for graphing as it immediately tells us key characteristics of the line.

• The slope \(m\) denotes the rate of change, or how steep the line is. For example, in \(y = -x + 1\), the slope is -1, indicating the line declines.
• The y-intercept \(b\) is the point where the line crosses the y-axis, such as ((0, 1)) for \(y = -x + 1\).

In the exercise, rewriting each equation into this form allows us to easily see how the lines look graphically and identify their slopes and intercepts. This approach efficiently communicates a line's behavior and position on the Cartesian plane.