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Chapter 1: Problem 61

Find the \(x\)-intercept and the \(y\)-intercept of the graph of the equation. \(3 x+y=1\)

Short Answer

Expert verified
The x-intercept of the graph is at \((1/3, 0)\) and the y-intercept is at \((0, 1)\).

Step by step solution

01

Find the x-intercept

Replace \(y\) in the equation with \(0\). We get: \(3x + 0 = 1\). Solving this equation for \(x\) gives us \(x = 1/3\). So, the x-intercept of the graph is at \((1/3, 0)\).
02

Find the y-intercept

Replace \(x\) in the equation with \(0\). We get: \(3*0 + y = 1\). Solving this equation for \(y\) gives us \(y = 1\). So, the y-intercept of the graph is at \((0, 1)\).

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Key Concepts

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

X-Intercept
The point where a graph crosses the x-axis is known as the x-intercept. It's a critical concept when dealing with linear equations and graphing, as it represents the value of x when y is zero.

To find it, we set y to zero because any point on the x-axis has a y-coordinate of zero. Consider the equation from our exercise, \(3 x + y = 1\). By substituting y with 0, we are left with \(3x = 1\), which simplifies to \(x = \frac{1}{3}\). Graphically, this means at the point \(\frac{1}{3}, 0\), the line touches the x-axis. This point is crucial in understanding the behavior of the line because it is one of the two intercepts that define its position on the graph.
Y-Intercept
Conversely, the y-intercept is where the graph intersects the y-axis. At this point, the value of x is zero, which aligns with a vertical line drawn from the origin to the point where the line intersects the y-axis.

In our example, we find the y-intercept by letting x be zero. Plugging it into the equation \(3 x + y = 1\) gives us \(y = 1\). The coordinate of the y-intercept is thus \(0, 1\), indicating the graph crosses the y-axis one unit above the origin. It is equally important as the x-intercept when plotting the line since these two points provide a quick method to sketch the graph of the equation.
Algebraic Solutions
The process we used to find the intercepts exemplifies how algebraic solutions help solve linear equations. By strategically replacing variables with zeros based on the intercept we want to find, we simplify the equation to just one variable.

This method doesn't just work for finding intercepts 鈥� it's a fundamental algebraic technique used to solve for unknowns whenever we have equations. For instance, if we had two linear equations, we could solve for both x and y by expressing one variable in terms of the other (substitution) or by balancing the equations to eliminate a variable (elimination method).
Linear Graph Intercepts
Together, linear graph intercepts offer a quick snapshot of a line鈥檚 layout on a coordinate plane. They represent specific points where the line crosses the axes. Knowing these points is like having a shortcut to sketch the entire line without plotting multiple individual points.

It's useful to note that the x- and y-intercepts are essentially the 'anchors' of the line. By drawing a straight line through them, you get the graph of the linear equation. The intercepts are especially helpful because you can find them algebraically without graphing the whole equation, saving time and providing an efficient way to understand the equation's graph.

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Most popular questions from this chapter

Compare each function with its parent function. State whether it contains a horizontal translation, vertical translation, both, or neither. Explain your reasoning. a. \(f(x)=2|x|-3\) b. \(f(x)=(x-8)^2\) c. \(f(x)=|x+2|+4\) d. \(f(x)=4 x^2\)

MODELING WITH MATHEMATICS The data pairs \((x, y)\) represent the average annual tuition \(y\) (in dollars) for public colleges in the United States \(x\) years after 2005 . Use the linear regression feature on a graphing calculator to find an equation of the line of best fit. Estimate the average annual tuition in 2020. Interpret the slope and \(y\)-intercept in this situation. (See Example 4.) $$ (0,11,386),(1,11,731),(2,11,848) $$ \((3,12,375),(4,12,804),(5,13,297)\)

Solve the system using the elimination method. \(2 x+y-z=9\) \(-x+6 y+2 z=-17\) \(5 x+7 y+z=4\)

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