Problem 10 Find the $x$ - and $y$ -inte... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 3: Problem 10

Find the $x$ - and $y$ -intercepts. $$ -x+y=0 $$

Short Answer

Expert verified

x-intercept: (0, 0); y-intercept: (0, 0).

Step by step solution

Identify equation type

The given equation is a linear equation in the standard form. An example of the standard form of a linear equation is $ax + by = c$. In this equation, $a = -1$, $b = 1$, and $c = 0$.

Find the x-intercept

To find the x-intercept, set $y = 0$ in the equation and solve for $x$. Substitute $y = 0$ into the equation:\[-x + 0 = 0\]Simplify and solve for $x$:\[-x = 0\]\[x = 0\]So, the x-intercept is $(0, 0)$.

Find the y-intercept

To find the y-intercept, set $x = 0$ in the equation and solve for $y$. Substitute $x = 0$ into the equation:\[-0 + y = 0\]Simplify and solve for $y$:\[y = 0\]So, the y-intercept is $(0, 0)$.

Conclusion

Both the x-intercept and y-intercept are at the origin, therefore the graph of the equation passes through the origin.

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Key Concepts

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

Understanding the x-intercept

The x-intercept of a line is the point where the line crosses the x-axis on a graph. At this point, the y-coordinate is always zero because the line has not yet moved up or down. To find the x-intercept in a given linear equation, you need to substitute zero for the y variable.
\[\text{Example:} -x+y=0\]

Set $y = 0$ in the equation.
The equation becomes $-x + 0 = 0$.
Simplify to find $x$.
In this case, $-x = 0$ leads to $x = 0$.

So, the x-intercept is the point $(0, 0)$. This point indicates where the line crosses the x-axis. In this example, it means the graph passes through the origin.

Understanding the y-intercept

The y-intercept of a line is where it crosses the y-axis. For this to occur, the x-coordinate must always be zero, since it's not moving left or right at that point. To find the y-intercept, set x equal to zero in your linear equation.
\[\text{Example:} -x+y=0\]

Set $x = 0$ in the equation.
The equation simplifies to $-0 + y = 0$ or simply $y = 0$.

This calculation indicates that the y-intercept is also $(0, 0)$. Like the x-intercept, the line of this equation also crosses the graph at the origin, showing that both intercept points coincide at the origin.

Equation Solving Techniques

Solving equations involves finding the values for the variables that make the equation true. When dealing with linear equations like $-x + y = 0$, it's all about finding values for x and y. What makes this equation special is the symmetry in its intercepts.

For seasoned equation solving:

Identify if it's a standard form linear equation: $ax + by = c$.
Notice the coefficients: here, $a = -1$, $b = 1$, $c = 0$.
Focus on isolating the variable, by getting it alone on one side of the equation.

For example, to find the x-intercept and y-intercept:

Set y and then x to zero, respectively, and solve for the other variable.
Recognize that both intercepts are at the origin $(0, 0)$, once substituted and simplified.

Ultimately, solving such a linear equation requires careful substitution and simplification, ensuring that each step logically follows the previous one to arrive at a solution. For practice, try substituting different values and notice the change in intercepts.

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91影视

Find the \(x\) - and \(y\) -intercepts. $$ -x+y=0 $$

Short Answer

Step by step solution

Identify equation type

Find the x-intercept

Find the y-intercept

Conclusion

Key Concepts

Understanding the x-intercept

Understanding the y-intercept

Equation Solving Techniques

One App. One Place for Learning.

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