Problem 17 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$x+y=5$$

Step-by-Step Solution

Answer

The solutions are \((0,5)\), \((2,3)\), and \((5,0)\).

Step 1: Understand the Equation

The given equation is a linear equation in two variables, which can be expressed in the form \(y = mx + c\). For \(x + y = 5\), rearrange it to get \(y = -x + 5\). This equation represents a line where every pair \((x, y)\) that satisfies the equation lies on the line.

Step 2: Choose Values for x

To find solutions, choose values for \(x\). Let's choose \(x = 0\), \(x = 2\), and \(x = 5\). These values will help us find corresponding values for \(y\) by substituting them into the equation.

Step 3: Calculate Corresponding y Values

Substitute the chosen \(x\) values into \(y = -x + 5\) to find \(y\).- For \(x = 0\):\[ y = -0 + 5 = 5 \]- For \(x = 2\):\[ y = -2 + 5 = 3 \]- For \(x = 5\):\[ y = -5 + 5 = 0 \]The solutions are \((0,5)\), \((2,3)\), and \((5,0)\).

Step 4: Draw the Graph

To plot the graph, draw a coordinate plane with x and y axes. Mark the points obtained from the equation: \((0,5)\), \((2,3)\), and \((5,0)\). These points should lie on a straight line. Connect the points to form the graph of the equation.

Key Concepts

Coordinate Plane

To solve many mathematical problems, the coordinate plane is a remarkably effective tool. It is a two-dimensional surface formed by two intersecting perpendicular lines, the x-axis and y-axis. These lines divide the plane into four quadrants, allowing us to locate and plot points using coordinate pairs \(x, y\). Each pair provides precise locations, helping us find solutions to equations like linear ones. Linear equations, when graphed, form a straight line on this plane.

Here are some essential features of the coordinate plane:

• The horizontal line is called the x-axis
• The vertical line is called the y-axis
• The intersection point of the axes is the origin, with coordinates \(0,0\)
• Points in the plane are described by ordered pairs \(x, y\)

Understanding how to navigate the coordinate plane is fundamental in locating the solutions of equations. When we solve a linear equation, we're essentially finding points on this plane that fall on a straight line, visually depicting the relationship between variables.

Graphing Solutions

Graphing solutions to equations involves plotting points on the coordinate plane to visualize how equations behave. The process begins by solving the equation for values of one variable, in this case, the y variable in terms of x. For the equation we have, \(x + y = 5\) rearranges to \(y = -x + 5\).

Here's how to graph solutions effectively:

• Decide on several x-values to calculate corresponding y-values.
• Substitute these x-values into the equation to determine each y-value.
• Write each solution as an ordered pair.
• Plot these points on the coordinate plane.
• Finally, connect the dots to illustrate the line.

By plotting points like \(0, 5\), \(2, 3\), and \(5, 0\), each solution represents a valid point from the linear equation. Connecting these points exhibits the behavior and direction of the linear graph. This visual representation helps validate the solutions and provides deeper insights into the relationship between variables within the equation.

Linear Graph

A linear graph represents solutions to a linear equation on the coordinate plane. Such a graph illustrates the relationship between two variables that produces a straight line. For the equation \(x + y = 5\), or \(y = -x + 5\), the linear graph visually shows every \(x\) and \(y\) pair that satisfies the equation, confirming that they lie on this straight line.

Key characteristics of linear graphs include:

• They form straight lines.
• Every point on the line is a solution to the equation.
• The slope of the line indicates its steepness and direction.
• The y-intercept (where the line crosses the y-axis) is an important feature, here it is 5.

These characteristics make linear graphs a valuable resource in visualizing relationships within equations. By setting different x-values and plotting the resulting points, one can create a clear and concise graph. Ultimately, this provides an immediate understanding of the linear relationship in question, aiding in analyses and predictions within various mathematical contexts.