Problem 24 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$4 x-2 y=4$$

Step-by-Step Solution

Answer

The three points are (0, -2), (1, 0), and (2, 2). Plot these to draw the line.

Step 1: Simplify the Equation

Start by simplifying the equation given in standard form, \(4x - 2y = 4\). Divide every term by 2 to simplify: \(2x - y = 2\). This represents a linear equation in two variables.

Step 2: Solve for y-intercept

To find the y-intercept, let \(x = 0\) in the equation \(2x - y = 2\). Substitute \(x = 0\) into the equation: \[2(0) - y = 2\] This simplifies to:\[-y = 2 \Rightarrow y = -2\]So, the y-intercept is \((0, -2)\).

Step 3: Solve for x-intercept

To find the x-intercept, let \(y = 0\) in the equation \(2x - y = 2\). Substitute \(y = 0\) into the equation:\[2x - (0) = 2\]This simplifies to:\[2x = 2 \Rightarrow x = 1\]So, the x-intercept is \((1, 0)\).

Step 4: Find a Third Point

For more accuracy, choose another value for \(x\) or \(y\) and solve for the other variable. Let's choose \(x = 2\):\[2(2) - y = 2\] Which simplifies to:\[4 - y = 2\]\[-y = -2\Rightarrow y = 2\]This gives the point \((2, 2)\).

Step 5: Draw the Graph

To graph the equation, plot the three points \((0, -2)\), \((1, 0)\), and \((2, 2)\) on a Cartesian plane. Draw a straight line through these points to represent the equation \(4x - 2y = 4\).

Key Concepts

Solving for Intercepts

Intercepts are essential for graphing linear equations. They show where the line crosses the x- and y-axes of the Cartesian plane. To find intercepts, you solve the equation for points where one variable is zero.

• Y-intercept: Set \(x = 0\) in the simplified linear equation \(2x - y = 2\). This means finding where the line crosses the y-axis. Substitute the value: \(2(0) - y = 2\) which simplifies to \(-y = 2\). This gives us \(y = -2\), so the y-intercept is \((0, -2)\).
• X-intercept: Set \(y = 0\) to find the x-intercept where the line crosses the x-axis. Substitute into the equation: \(2x - 0 = 2\), simplifying to \(2x = 2\), leading to \(x = 1\). Therefore, the x-intercept is \((1, 0)\).

These intercepts help sketch the line of the equation on the graph.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing a line through them. This makes the relationship between x and y visible and understandable. First, find points on the line using intercepts and any other solutions you can compute. For the equation \(2x - y = 2\),

• We plotted the y-intercept \((0, -2)\) and the x-intercept \((1, 0)\).
• To ensure the accuracy of the graph, another point, such as \((2, 2)\), is useful. This point was found by substituting \(x = 2\) into the equation to solve for y. The result was: \(4 - y = 2\), simplifying to \(y = 2\).

Once these points are plotted, draw a straight line through them. This line is the graphical representation of the equation.

Cartesian Plane

The Cartesian plane is a two-dimensional surface for graphing equations. It consists of a horizontal x-axis and a vertical y-axis that cross each other at the origin \((0, 0)\). Every point on this plane is represented by coordinates \((x, y)\), telling you how far along the horizontal and vertical axis the point lies.

• X-axis: The horizontal axis, where the value of y is zero.
• Y-axis: The vertical axis, where the value of x is zero.
• Origin: Where the x-axis and y-axis meet, with coordinates \((0, 0)\).

Plotting points on the Cartesian plane helps visualize solutions from linear equations. As you plot each point for the equation \(4x - 2y = 4\), like \((0, -2)\), \((1, 0)\), and \((2, 2)\), you'll see the structure of the linear relationship unfold. This makes understanding equations not just theoretical but also visual.