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The x-intercept is a fundamental concept in graphing and understanding linear equations. To locate the x-intercept of a given linear equation, you set the value of the dependent variable, or the output variable—which is usually "y"—to zero. In simpler terms, the x-intercept is the location where the graph of the equation crosses, or touches, the x-axis without going above or below the axis. This point will always be in the form

• \((x, 0)\)

Thus, for the equation \(15x + 4y = -60\), setting \(y\) to zero and solving for \(x\) gets us \(x = -4\), resulting in the x-intercept at point

• \((-4, 0)\).

This tells us that when \(y\) is zero, \(x\) is

• \(-4\), marking the intersection with the x-axis.

Recognizing this point helps in quickly sketching out

• a graph of the line on a coordinate plane.

The y-intercept of a linear equation is the point where the line crosses the y-axis. Similar to finding the x-intercept, the y-intercept is found by setting \(x = 0\). This point is crucial in understanding how a linear equation behaves and looks on a graph. The y-intercept is expressed in a coordinate form as

• \((0, y)\)

For the equation \(15x + 4y = -60\), when we set \(x\) to zero and solve for \(y\), we find that \(y = -15\). This gives us the y-intercept at the point

• \((0, -15)\).

At this intersection, the line meets the y-axis. This specific point provides a starting reference that mathematically represents the vertical intercept of the line. Determining and plotting the y-intercept is essential in accurately drawing the complete line of any given linear equation.

Graphing linear equations is the process of visually representing the solution set of an equation on a coordinate plane. When we say we want to graph the equation \(15x+4y=-60\), we're looking to draw a straight line. To correctly graph this linear equation:
Identify both the x-intercept and y-intercept, which in this problem are \((-4, 0)\) and \((0, -15)\) respectively.

• Use these intercepts as reference points.
• Plot both points on the coordinate plane.
• Draw a straight line that passes through both plotted intercepts.

This line will graphically represent all solutions to the original equation. Each point on this line is a solution, demonstrating a balance between \(x\) and \(y\) that satisfies the equation. Understanding how to locate the intercepts first simplifies graphing by providing clear reference markers.

The coordinate plane is a two-dimensional surface defined by a horizontal line (x-axis) and a vertical line (y-axis), which intersect at a point called the origin

• \((0, 0)\).

This plane is crucial for graphing equations as it provides a framework structure with infinite possible points by which graphs are drawn. In graphing, each point is defined by an ordered pair of numbers, such as

• \((x, y)\).

The coordinate plane is divided into four quadrants:

• First Quadrant: Both \(x\) and \(y\) are positive.
• Second Quadrant: \(x\) is negative, \(y\) is positive.
• Third Quadrant: Both \(x\) and \(y\) are negative.
• Fourth Quadrant: \(x\) is positive, \(y\) is negative.

When graphing a linear equation, such as our earlier example, your goal is to correctly identify where the line should be placed within these quadrants. For instance, the equation \(15x+4y=-60\) has intercepts that place it crossing from quadrant two to quadrant four. Mastery of moving through and understanding the coordinate plane is essential for accurate mathematical graphing.