Problem 20 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$x+y=3$$

Step-by-Step Solution

Answer

Three solutions are \((0, 3)\), \((1, 2)\), and \((2, 1)\). Plotting these points will create a straight line graph representing the equation \(x + y = 3\).

Step 1: Understand the Equation

The equation given is a linear equation in two variables, "x" and "y." Specifically, it is in the form \(x + y = 3\), which is a straight line. We need to find three points \((x, y)\) that satisfy this equation.

Step 2: Find the First Solution

To find a point that satisfies the equation, you can choose a value for \(x\) and then solve for \(y\). Let's choose \(x = 0\). Substitute \(x = 0\) into the equation:\[ 0 + y = 3 \]Solving for \(y\) gives:\[ y = 3 \]So, the first solution is \((0, 3)\).

Step 3: Find the Second Solution

Next, let's choose \(x = 1\). Substitute \(x = 1\) into the equation:\[ 1 + y = 3 \]Solving for \(y\) gives:\[ y = 2 \]Thus, the second solution is \((1, 2)\).

Step 4: Find the Third Solution

For the third solution, let's choose \(x = 2\). Substitute \(x = 2\) into the equation:\[ 2 + y = 3 \]Solving for \(y\) gives:\[ y = 1 \]Therefore, the third solution is \((2, 1)\).

Step 5: Describe Graph Plot

The three solutions found are \((0, 3)\), \((1, 2)\), and \((2, 1)\). These points lie on the line represented by the equation \(x + y = 3\). By plotting these points on a graph and drawing a straight line through them, you can visualize the entire line, which shows all solutions to the equation.

Key Concepts

Linear Equations

A linear equation is an algebraic expression that creates a straight line when graphed on a coordinate plane. It typically has two variables, often referred to as "x" and "y." For example, the equation \(x + y = 3\) is a linear equation. Linear equations are also known for their simplicity and pattern; they describe a relationship between the input (usually "x") and the output (usually "y") that is constant. This means the equation follows a straight-line path.

When you look at \(x + y = 3\), you can see that to find different solutions, you select values for "x" and solve for "y" to satisfy the equation. This equation does not change no matter which "x" value you choose. That's the beauty of linear equations—once you understand the pattern they follow, predicting values and graphing becomes easy.

Coordinate Plane

The coordinate plane is a flat surface created by two lines, the x-axis, and the y-axis, intersecting at a point called the origin. The origin is labeled \(0, 0\). You use the coordinate plane to visually represent mathematical equations such as linear equations.

Each axis represents a variable from the equation, with the x-axis typically representing "x" and the y-axis representing "y." This system allows you to plot points that are solutions to an equation. So, when you have points like \(0, 3\), \(1, 2\), and \(2, 1\) from our equation \(x + y = 3\), they are plotted on this plane. You move horizontally based on the x-value and vertically based on the y-value. Connecting these points with a line will help visualize the complete set of solutions for the equation. It turns the abstract numbers into a clear picture of relationships.

Solving Equations with Two Variables

Solving equations with two variables means finding all pairs of values that make the equation true. With linear equations like \(x + y = 3\), there are infinite possible solutions because you can select various "x" values and compute the corresponding "y" values to satisfy the equation.

Here's how you can solve it step by step:

• Choose a value for "x".
• Substitute the "x" value into the equation.
• Solve the resulting equation for "y".
• Record the pair \(x, y\).

For instance, substituting \(x = 1\) into the equation \(x + y = 3\), solving yields \(y = 2\), giving the point \(1, 2\). Repeat this with different x-values to find other points.

The primary goal is to connect these solutions in a systematic pattern, presented easily by plotting them on the coordinate plane. This graph demonstrates both the simplicity and versatility of solutions in linear equations.