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To solve a system of linear equations by graphing, it's helpful to rewrite each equation in what's known as the slope-intercept form. This is expressed as the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept. Adjusting equations into this format makes plotting them on a graph straightforward. For instance, given the equation 2x + y = -1, subtract 2x from both sides to obtain y = -2x - 1. Now, it's in slope-intercept form with a slope \( m \) of -2, and a y-intercept \( b \) of -1. Incidentally, making similarly simple transformations to the second equation \( y - x = 5 \) yields y = x + 5. Now both equations are in slope-intercept form, and you are ready to move to the next steps.

The y-intercept of an equation is the point where the line crosses the y-axis, and it occurs when x equals 0. Identifying the y-intercept in the slope-intercept form is easy; it’s simply the constant \( b \) in \( y = mx + b \). For example, in the equation \( y = -2x - 1 \), the y-intercept is -1, meaning the line crosses the y-axis at (0, -1). For the equation \( y = x + 5 \), the y-intercept is 5, indicating a crossing at (0, 5). These points are critical as they provide a starting point to plot the lines on the graph.

Once the y-intercepts are identified, you can proceed with plotting points. Start by marking the y-intercepts on the graph, such as (0, -1) and (0, 5). Next, use the slopes to find additional points. For \( y = -2x - 1 \), the slope of -2 means moving down 2 units for every unit moved right. Starting at (0, -1) and applying this method leads to the next point (1, -3). Similarly, for the line \( y = x + 5 \), a slope of 1 means moving up 1 unit for every unit moved right. Thus, from (0, 5) move up to (1, 6). Plotting these points helps to establish the path of the line.

Working with linear equations involves understanding their properties and behavior when graphed. Each linear equation forms a straight line when plotted on a coordinate plane. The relationship within a linear equation signifies a constant rate of change, represented graphically by the slope. For instance, a slope of 1 means for each unit increase in x, y also increases by 1. In contrast, a slope of -2 indicates y decreases by 2 units with each unit increase in x. Recognizing these properties facilitates accurate graphing and solution determination for systems of linear equations.

Graphing lines from linear equations entails plotting points derived from the slope and y-intercept, then drawing a straight line through these points. For example, starting from the y-intercept (0, -1) for \( y = -2x - 1 \), use the slope -2 to find another point, such as (1, -3). Do similarly for \( y = x + 5 \) with points (0, 5) and (1, 6). Once points for each are plotted, draw lines through them. The intersection of these lines marks the solution to the system, representing values of x and y satisfying both equations simultaneously. This visual approach simplifies solving the system.