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Graphing linear equations is a foundational concept in algebra. It involves drawing a line on the coordinate plane that represents all solutions of a given linear equation. A linear equation forms a straight line when graphed. For example, the equation \(y=-\frac{1}{4}x\) creates a line where the slope, or steepness, is \(-\frac{1}{4}\). To graph a linear equation, you need to plot points that satisfy the equation. These points are derived by choosing values for \(x\) and calculating the corresponding \(y\) values. The line you draw through these points represents all possible solutions, extending infinitely in both directions on the graph. Moreover, understanding the intercepts鈥攚here the line crosses the x-axis and y-axis鈥攃an be extremely helpful. In \(y=-\frac{1}{4}x\), the line crosses the origin (0,0) because there is no constant term.

Solutions of linear equations are the pairs of \((x, y)\) that make the equation true. For the given equation \(y=-\frac{1}{4}x\), each solution creates a point on the graph. In other words, if you substitute an \(x\) value and the corresponding \(y\) value into the equation, it should hold true.In the step-by-step solution, we identified five solutions by selecting values for \(x\) such as -2, 0, 2, 4, and 6. By substituting these into our equation, we calculated the respective \(y\) values which are 0.5, 0, -0.5, -1, and -1.5. All these points are solutions because they maintain equality in the equation.Finding solutions is critical when graphing as it provides points to plot, confirming the line's correctness. It's essential to check multiple points, at least three, to ensure accuracy and that the points indeed lie on a straight line.

A table of values is a useful tool for organizing and calculating coordinates to graph a linear equation. It typically consists of two columns, one for \(x\) values and one for \(y\) values, derived from substituting \(x\) into the linear equation.When constructing a table of values, select a range of \(x\) values. It helps to choose both negative and positive numbers to see how the graph behaves across different regions. For \(y=-\frac{1}{4}x\), we chose \(-2, 0, 2, 4,\) and \(6\) as our \(x\) values. After computation, the respective \(y\) values added to the table were 0.5, 0, -0.5, -1, and -1.5.Using a table of values provides a clear systematic approach to finding several solutions to graph accurately. This table not only assists in graphing but also offers a visual representation of how \(x\) changes affect \(y\) in the equation.

The coordinate plane is a two-dimensional surface where we can graph points, lines, and shapes. This plane is divided into four quadrants by the x-axis and y-axis intersecting at the origin (0,0).In graphing a linear equation like \(y=-\frac{1}{4}x\), we use the coordinate plane as the canvas. Each point from our table of values corresponds to an ordered pair \((x, y)\) that we plot on this plane. The x-axis represents horizontal positions, while the y-axis represents vertical. It's vital to properly scale your axes to encompass all relevant points from your table of values.Once the points are plotted, you connect them with a straight line. This line represents all solutions of the equation on the coordinate plane. The clarity of this visual representation is why mastering the coordinate plane is crucial in graphing linear equations.