Problem 101 - Elementary Algebra

Question

Construct the graph of \(y=-2 x+4\).

Step-by-Step Solution

Answer

Answer: The two points used to construct the graph of the linear equation \(y = -2x + 4\) are (0, 4) and (2, 0). They are found by plugging in x = 0 and x = 2 into the equation and calculating the corresponding y-values.

Step 1: Find the first point on the graph

Plug in x = 0 into the equation \(y = -2x + 4\): \(y = -2(0) + 4\) \(y = 4\) The first point on the graph is (0, 4).

Step 2: Find the second point on the graph

Plug in x = 2 into the equation \(y = -2x + 4\): \(y = -2(2) + 4\) \(y = -4 + 4\) \(y = 0\) The second point on the graph is (2, 0).

Step 3: Plot the two points

Now, plot the two points (0, 4) and (2, 0) on a coordinate plane.

Step 4: Draw a straight line through the points

Finally, draw a straight line through the two points. This line is the graph of the linear equation \(y = -2x + 4\).

Key Concepts

Coordinate Plane

Imagine a blank sheet with a big plus sign drawn on it. This plus sign is actually the basis of what we call a coordinate plane, a two-dimensional surface where we can accurately locate points using pairs of numbers. The horizontal line is known as the x-axis, and the vertical line is the y-axis. These axes intersect at a point called the origin, which has the coordinates (0,0).

When graphing linear equations, the coordinate plane allows us to show every combination of x and y that makes the equation true as a point on this plane. By plotting a few of these points and connecting them, we can visualize the solution as a line. This visual representation can be incredibly helpful in understanding how the equation behaves across different values.

Linear Equation

A linear equation is a straight line when graphed on a coordinate plane. It's an algebraic equation in which each term is either a constant or the product of a constant and a single variable. A linear equation in two variables, like x and y, has an infinite number of solutions that form a straight line.

For instance, in the equation y = -2x + 4, the -2 and 4 are constants, with -2 being the coefficient of x, the variable. The solutions to this equation are all the points through which the line passes. Plotting these points and connecting them gives us the visual solution to the equation.

Slope-Intercept Form

One popular way of expressing linear equations is the slope-intercept form, which is written as y = mx + b. Here, m represents the slope of the line, and b is the y-intercept, which is where the line crosses the y-axis.

In the equation from our exercise, y = -2x + 4, the slope is -2 and the y-intercept is 4. This means the line goes down two units for every one unit it moves to the right, which is why it's a negative slope. The line intersects the y-axis at the point (0,4). Understanding this format makes it much easier to graph linear equations since you can quickly identify the slope and starting point of the line on the graph.