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Magazine Chapter 1: Problem 73
Find the x- and y-intercepts of the graph of the equation. (a) \(9 x^{2}-4 y^{2}=36\) (b) \(y-2 x y+4 x=1\)
Short Answer
Expert verified
(a) x-intercepts: (2, 0), (-2, 0); no y-intercepts; (b) x-intercept: (1/4, 0); y-intercept: (0, 1).
Step by step solution
01
Understanding x-intercepts for equation (a)
To find the x-intercepts of the equation \(9x^2 - 4y^2 = 36\), set \(y = 0\) and solve for \(x\). The x-intercepts occur at the points where the graph crosses the x-axis.
02
Solving for x-intercepts of equation (a)
Substitute \(y = 0\) into the equation: \(9x^2 - 4(0)^2 = 36\). This simplifies to \(9x^2 = 36\). Solving for \(x\) gives \(x^2 = 4\), so \(x = 2\) or \(x = -2\). Thus, the x-intercepts are \((2, 0)\) and \((-2, 0)\).
03
Understanding y-intercepts for equation (a)
To find the y-intercepts, set \(x = 0\) and solve for \(y\). The y-intercepts occur where the graph crosses the y-axis.
04
Solving for y-intercepts of equation (a)
Substitute \(x = 0\) into the equation \(9(0)^2 - 4y^2 = 36\), which simplifies to \(-4y^2 = 36\). Solving for \(y\) we get \(y^2 = -9\), which has no real solution. Therefore, there are no y-intercepts for equation (a).
05
Understanding x-intercepts for equation (b)
For the equation \(y - 2xy + 4x = 1\), set \(y = 0\) and solve for \(x\) to find the x-intercepts.
06
Solving for x-intercepts of equation (b)
Substitute \(y = 0\) into the equation \(0 - 2x(0) + 4x = 1\), simplifying to \(4x = 1\). Solving for \(x\) gives \(x = \frac{1}{4}\). Therefore, the x-intercept is \((\frac{1}{4}, 0)\).
07
Understanding y-intercepts for equation (b)
To find the y-intercepts for the equation \(y - 2xy + 4x = 1\), set \(x = 0\) and solve for \(y\).
08
Solving for y-intercepts of equation (b)
Substitute \(x = 0\) into the equation \(y - 2(0)y + 4(0) = 1\), which simplifies to \(y = 1\). Therefore, the y-intercept is \((0, 1)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
x-intercepts
In mathematics, the x-intercept of a graph is where the graph crosses the x-axis. This happens when the y-value is zero. To find the x-intercepts of an equation, set the variable for the y-axis to zero and solve for the x variable. This efficient method helps identify the roots or solutions of an equation in terms of x.
For example, consider the equation from the original problem:
For example, consider the equation from the original problem:
- (a) For the equation \( 9x^2 - 4y^2 = 36 \), setting \( y = 0 \) results in solving \( 9x^2 = 36 \), which gives the x-intercepts \((2, 0)\) and \((-2, 0)\).
- (b) For the equation \( y - 2xy + 4x = 1 \), setting \( y = 0 \) leads to solving \( 4x = 1 \), resulting in the x-intercept \((\frac{1}{4}, 0)\).
y-intercepts
The y-intercept is the point where a graph crosses the y-axis. In this case, the x-value is zero because points on the y-axis have no horizontal (x) movement. By setting \( x = 0 \) in an equation, you solve for the y variable to find the y-intercepts. This simple change in the equation unveils vital vertical information about the graph.
Looking at the original problem:
Looking at the original problem:
- (a) For the equation \( 9x^2 - 4y^2 = 36 \), setting \( x = 0 \) results in \( -4y^2 = 36 \). After solving, this shows there are no real y-intercepts because it leads to nonsensical solutions in real numbers.
- (b) In the equation \( y - 2xy + 4x = 1 \), setting \( x = 0 \) leads to the equation \( y = 1 \), giving the y-intercept \((0, 1)\).
solving equations
Solving equations is a key process in algebra and calculus. It involves finding the values of variables that satisfy a given equation. There are numerous techniques to solve equations, such as factoring, substitution, and using graphical methods.
Equations like those encountered in the exercise can include quadratic or linear types:
Equations like those encountered in the exercise can include quadratic or linear types:
- Quadratic Equations: An equation like \( 9x^2 - 4y^2 = 36 \) involves quadratic terms, which can have two potential solutions. Quadratics may require setting one variable to zero and rearranging it to solve for others, as shown for the x-intercepts in equation (a).
- Linear Equations: Equations like \( y - 2xy + 4x = 1 \) can often be simplified to find solutions easily, especially when setting one variable to zero, providing intercepts as direct solutions to variable adjustments.
