Problem 2 - Algebra and Trigonometry

Question

(a) To find the \(x\) -intercept(s) of the graph of an equation, we set _____ equal to 0 and solve for _____ So the \(x\) -intercept of \(2 y=x+1\) is _____. (b) To find the \(y\) -intercept(s) of the graph of an equation, we set _____ equal to 0 and solve for _____ So the \(y\) -intercept of \(2 y=x+1\) is _____.

Step-by-Step Solution

Answer

(a) Set \(y = 0\). \(x = -1\). (b) Set \(x = 0\). \(y = \frac{1}{2}\).

Step 1: Understand Equation Structure

The given equation is presented in the form of a linear equation. Rewriting it, we have the equation \(2y = x + 1\).

Step 1: Finding x-Intercepts

To find the \(x\)-intercept, we set \(y = 0\) and solve for \(x\). Substitute \(y = 0\) into the equation \(2y = x + 1\), resulting in \(2(0) = x + 1\). This simplifies to \(x + 1 = 0\). Solving for \(x\), we get \(x = -1\). Therefore, the \(x\)-intercept is \((-1, 0)\).

Step 2: Finding y-Intercepts

To find the \(y\)-intercept, we set \(x = 0\) and solve for \(y\). Substitute \(x = 0\) into the equation \(2y = x + 1\), which gives \(2y = 0 + 1\). Simplifying, we find \(2y = 1\). Solving for \(y\), we divide both sides by 2, resulting in \(y = \frac{1}{2}\). Therefore, the \(y\)-intercept is \((0, \frac{1}{2})\).

Key Concepts

x-intercepts

The x-intercept of a graph is a point where the graph crosses the x-axis. At this point, the y-coordinate is always zero. To find the x-intercept of an equation, we substitute zero for the y variable and solve for x. In our given equation, which is transformed into the form \(2y = x + 1\), finding the x-intercept requires setting \(y = 0\). This substitution transforms the equation into \(0 = x + 1\). Solving for x, subtract 1 from both sides to get \(x = -1\). Thus, the x-intercept for this linear equation is the point \((-1, 0)\). Thinking about an x-intercept is like asking, "At what point does the graph hit the floor of the coordinate grid?"

• Set y to zero.
• Solve the resulting equation for x.
• The solution gives you the x-coordinate of the intercept.

y-intercepts

Similarly, the y-intercept is the point where the graph crosses the y-axis. At this coordinate, the x value is zero. To find the y-intercept, substitute zero for the x variable, and solve the resulting equation for y. In the equation \(2y = x + 1\), when we substitute \(x = 0\), the equation simplifies to \(2y = 1\). Solving for y involves dividing both sides by 2, which results in \(y = \frac{1}{2}\). Therefore, the y-intercept is located at \((0, \frac{1}{2})\).

• Set x to zero.
• Solve the resulting equation to find y.
• The value you get is the y-coordinate of the intercept.

The y-intercept answers the question: "Where does the graph climb upwards and touch the y-axis?"

solving equations

Solving equations successfully involves manipulating them to find the value of the variable. In linear equations, you'll typically see one variable on each side of the equation.

A common approach to solve linear equations is to isolate the variable you wish to solve for. This generally involves the following steps:

• Start by identifying which variable you need to solve for, based on what intercept you are looking for.
• Simplify each side of the equation if needed, by performing operations such as combining like terms or distributing.
• To isolate your variable, use inverse operations: add or subtract to clear constants, and multiply or divide to clear coefficients.
• Always keep your equation balanced by performing the same operations on both sides.

In our example equations, to find the x-intercept, we needed to make y zero and solve for x, which meant simple addition or subtraction. For the y-intercept, setting x to zero and solving for y required us to divide, since the variable y had a coefficient beside it. Understanding these basics is essential to seamlessly progress through any linear equation task.