Problem 2 (a) To find the \(x\) -intercept... [FREE SOLUTION]

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Chapter 2: Problem 2

(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 ___.

Short Answer

Expert verified

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

Step by step solution

Understand Equation Structure

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

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 \((-1, 0)\).

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 \((0, \frac{1}{2})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

(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 ___.

Short Answer

Step by step solution

Understand Equation Structure

Finding x-Intercepts

Finding y-Intercepts

Key Concepts

x-intercepts

y-intercepts

solving equations

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