Finding Graph Intercepts on Coordinate Axes

The question asks us to find the points where the graph of the linear equation $\text{57x - 19y = 399}$ intersects the coordinate axes. The coordinate axes are the x-axis and the y-axis.

A point where a graph intersects the x-axis is called the x-intercept. At any point on the x-axis, the y-coordinate is always 0.

A point where a graph intersects the y-axis is called the y-intercept. At any point on the y-axis, the x-coordinate is always 0.

Calculating the X-intercept

To find the x-intercept, we set $\text{y = 0}$ in the given equation and solve for $\text{x}$.

The equation is: $\text{57x - 19y = 399}$

Substitute $\text{y = 0}$:

$\text{57x - 19(0) = 399}$

$\text{57x - 0 = 399}$

$\text{57x = 399}$

Now, solve for $\text{x}$ by dividing both sides by 57:

$\text{x = } \frac{399}{57}$

We can perform the division:

So, $\text{x = 7}$.

The x-intercept occurs when $\text{x = 7}$ and $\text{y = 0}$. The coordinates of the x-intercept are $(\text{7, 0})$.

Calculating the Y-intercept

To find the y-intercept, we set $\text{x = 0}$ in the given equation and solve for $\text{y}$.

The equation is: $\text{57x - 19y = 399}$

Substitute $\text{x = 0}$:

$\text{57(0) - 19y = 399}$

$\text{0 - 19y = 399}$

$\text{-19y = 399}$

Now, solve for $\text{y}$ by dividing both sides by -19:

$\text{y = } \frac{399}{-19}$

We can perform the division:

So, $\text{y = -21}$.

The y-intercept occurs when $\text{x = 0}$ and $\text{y = -21}$. The coordinates of the y-intercept are $(\text{0, -21})$.

Summary of Intercepts

The graph of the equation $\text{57x - 19y = 399}$ cuts the coordinate axes at the following points:

• On the x-axis: $(\text{7, 0})$
• On the y-axis: $(\text{0, -21})$

Comparing these results with the given options, we find that the graph cuts the x-axis at $(\text{7, 0})$ and the y-axis at $(\text{0, -21})$.

Revision Table: Graph Intercepts

Additional Information: Linear Equation Graphs

A linear equation in two variables, like $\text{57x - 19y = 399}$, represents a straight line when graphed on a coordinate plane.

• The general form of a linear equation is $\text{Ax + By = C}$, where A, B, and C are constants, and A and B are not both zero.
• In our equation, $\text{A = 57}$, $\text{B = -19}$, and $\text{C = 399}$.
• The intercepts are often useful points for quickly sketching the graph of a linear equation. Once you have the two intercepts, you can draw a straight line through them.
• Every point $(\text{x, y})$ that lies on the line satisfies the equation.
• A non-vertical line (like this one) will have exactly one x-intercept and exactly one y-intercept, unless it passes through the origin (0,0), in which case both intercepts are (0,0).