Answered

Graph the linear equation by finding and plotting its intercepts.

$5x - 3y = 15$



Solve for $x$:

$\begin{aligned}

5x - 3y &= 15 \\

5x - 3(0) &= 15 \quad \text{Let } y = 0. \\

5x &= 15 \quad \text{Simplify.} \\

x &= 3 \quad \text{Divide.}

\end{aligned}$



The $x$-intercept is $(3, 0)$. The graph of $5x - 3y = 15$ passes through the point $(3, 0)$.



A $y$-intercept of a graph is a point where the graph intersects the $y$-axis. To find the $y$-intercept, let $x = 0$ and solve for $y$:

$\begin{aligned}

5x - 3y &= 15 \\

5(0) - 3y &= 15 \quad \text{Let } x = 0. \\

-3y &= 15 \quad \text{Simplify.} \\

y &= \square \quad \text{Divide.}

\end{aligned}$

Verified Answer :

  • Set the variable \(x\) to \(0\) to find the y-intercept of the equation \(5x - 3y = 15\).
  • Simplify the resulting equation to \(-3y = 15\).
  • Divide by \(-3\) to find that \(y = -5\).
  • The final y-intercept is \((0, -5)\), so the value for y is \(\boxed{-5}\).

Explanation

  1. Understanding the Objective

To find the y-intercept of the linear equation \(5x - 3y = 15\), we identify the point where the graph crosses the y-axis. At this point, the x-coordinate must be zero. By substituting \(x = 0\) into the equation, we can solve for the corresponding y-value.

  1. Setting up the Equation

Start with the given linear equation:
\[5x - 3y = 15\]
Substitute \(x = 0\) into the equation to isolate the y-term:
\[5(0) - 3y = 15\]
Simplify the expression by evaluating \(5 \times 0 = 0\):
\[-3y = 15\]

  1. Solving for the y-intercept

To solve for \(y\), divide both sides of the equation by \(-3\):
\[y = \frac{15}{-3}\]
Performing the division gives:
\[y = -5\]
Thus, the y-intercept is the point \((0, -5)\).

Examples

In real life, linear equations are used to model relationships with a constant rate of change. For example, if you are filling a pool, the amount of water in the pool over time can be modeled by a linear equation. The y-intercept represents the initial amount of water in the pool before you started filling it (at time \(x = 0\)). Understanding how to find intercepts helps you quickly determine starting values and rates of change in many practical situations, like budgeting, travel, or physics.

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