Question

Find the \(x\) -intercept(s) and \(y\) -intercept(s) of the graph of \(3 x-8 y=6\) Find the \(x\) -intercept(s) and \(y\) -intercept(s) of the graph of \(3 x-8 y=6\)

Short answer

x-intercept: (2, 0), y-intercept: (0, -3/4)

Step-by-step solution

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Starting with the equation: \[3x - 8y = 6\], Substitute y = 0: \[3x - 8(0) = 6\] This simplifies to: \[3x = 6\]. Now, divide both sides by 3: \[x = 2\]. So, the x-intercept is \((2, 0)\).

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Starting with the same equation: \[3x - 8y = 6\], Substitute x = 0: \[3(0) - 8y = 6\] This simplifies to: \[-8y = 6\]. Now, divide both sides by -8: \[y = -\frac{6}{8} = -\frac{3}{4}\]. So, the y-intercept is \((0, -\frac{3}{4})\).

Key concepts

x-intercept

To identify the x-intercept of a linear equation, you must set the y-variable to zero and solve for x. This involves isolating x on one side of the equation.

For example, in the equation \(3x - 8y = 6\), follow these steps to find the x-intercept:

• Replace y with 0 in the equation.
• The equation simplifies to \(3x = 6\).
• Solve for x by dividing both sides by 3.
• Thus, the solution is \(x = 2\).

Therefore, the x-intercept is the point \((2, 0)\). This is the point where the graph crosses the x-axis.

y-intercept

Identifying the y-intercept in a linear equation requires setting the x-variable to zero and solving for y. This simplifies the equation to solve for the y-intercept's value.

For instance, using the same equation \(3x - 8y = 6\), find the y-intercept as follows:

• Substitute x with 0 in the equation.
• The equation simplifies to \(-8y = 6\).
• Solve for y by dividing both sides by -8.
• This yields \(y = -\frac{3}{4}\).

Consequently, the y-intercept is the point \((0, -\frac{3}{4})\). This is where the graph intersects the y-axis.

solving linear equations

Solving a linear equation involves finding the values of variables that satisfy the equation. This process usually includes isolating the variable of interest on one side of the equation.

Let's break down the steps using the example \(3x - 8y = 6\):

• Identify what you're solving for (x or y).
• If solving for x, set y to 0 and solve for x; if solving for y, set x to 0 and solve for y.
• Simplify the equation to isolate the variable.
• For x, divide by the coefficient of x; for y, divide by the coefficient of y.

Through these steps, you can find specific points where the equation intersects the axes, known as intercepts. Mastering this helps in analyzing and graphing linear equations accurately.