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In the realm of linear equations, the y-intercept is a fundamental concept that helps in understanding the relationship between variables. The y-intercept is the point where a line crosses the y-axis of a graph. This occurs when the value of \( x \) is equal to zero.
To find the y-intercept of the equation \( y = 5x \), we set \( x = 0 \). By substituting \( x = 0 \) into the equation, we get:

• \( y = 5(0) \)
• \( y = 0 \)

Therefore, the y-intercept of the equation \( y = 5x \) is \((0, 0)\). This means the line crosses the y-axis at the origin. Understanding the y-intercept is crucial when plotting linear equations and provides a clear starting point for drawing the graph.

The x-intercept is another crucial element of linear equations, representing the point where the line crosses the x-axis. Finding the x-intercept requires setting the value of \( y \) to zero and solving for \( x \).
For the equation \( y = 5x \), setting \( y = 0 \) and solving gives:

• \( 0 = 5x \)
• \( x = 0 \)

Hence, the x-intercept is also at \((0, 0)\). This indicates that not only does the line pass through the y-intercept at the origin, but it also crosses the x-axis at this point. The x-intercept is valuable for understanding the line's behavior and for confirming its pathway on the graph.

Graphing linear equations vividly displays the relationship between variables, allowing us to visualize their interaction. In the equation \( y = 5x \), both the y- and x-intercepts are at the origin \((0, 0)\). To draw the graph, plot this shared intercept first.
Since it's essential to identify at least another point to construct the line, choose a different \( x \) value. For example, if we set \( x = 1 \), we find:

• \( y = 5(1) \)
• \( y = 5 \)

This gives us the point \((1, 5)\). Now, plot this point on the graph. With the points \((0, 0)\) and \((1, 5)\), draw a straight line that extends through both points. This line represents the equation \( y = 5x \). This graphical representation demonstrates that as \( x \) increases by one, \( y \) increases by five, highlighting the equation's slope and making the linear relationship tangible.