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Chapter 2: Problem 56

Find the \(x\) - and \(y\) -intercepts. $$ -3 x-5 y=60 $$

Short Answer

Expert verified
The x-intercept is (-20, 0) and the y-intercept is (0, -12).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation. Then solve for x.\(-3x - 5(0) = 60\)\(-3x = 60\)Divide both sides by -3:\(x = \frac{60}{-3}\)\(x = -20\)Therefore, the x-intercept is (-20, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation. Then solve for y.\(-3(0) - 5y = 60\)\(-5y = 60\)Divide both sides by -5:\(y = \frac{60}{-5}\)\(y = -12\)Therefore, the y-intercept is (0, -12).

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Key Concepts

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

x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. This happens when the y-value is zero. To find the x-intercept, we set the y-value to zero and solve for x in the equation. Let's look at our given equation: $$-3x - 5y = 60$$ Set y to 0: $$-3x - 5(0) = 60$$ Simplifying: $$-3x = 60$$ Now, divide by -3 to isolate x: $$ x = \frac{60}{-3}$$ So, $$ x = -20$$ The x-intercept of the line is at (-20, 0). This means the line crosses the x-axis at this point.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. This occurs when the x-value is zero. To find the y-intercept, we set the x-value to zero and solve for y. Using our equation: $$-3x - 5y = 60$$ Set x to 0: $$-3(0) - 5y = 60$$ Simplify: $$-5y = 60$$ Now, divide by -5 to isolate y: $$ y = \frac{60}{-5}$$ Thus, $$ y = -12$$ The y-intercept is (0, -12), which shows that the line crosses the y-axis at this point.
solving linear equations
Solving linear equations means finding the values of the variables that make the equation true. In our example, the equation is linear because it can be graphed as a straight line. Let's break down the steps:
  • Identify the equation: $$-3x - 5y = 60$$.
  • Decide which variable to solve for first. For intercepts, we set one variable to zero.
  • Solve for the remaining variable.
  • For the x-intercept, set $$y = 0$$ and solve for $$x$$.
  • For the y-intercept, set $$x = 0$$ and solve for $$y$$.
These methods help us find the points where the line crosses the axes.
coordinate geometry
Coordinate geometry, also known as analytic geometry, involves describing geometric figures using coordinates on a plane. Here, we're using it to find where our line intersects the x- and y-axes. Coordinate geometry combines algebra and geometry:
  • The x-coordinate describes the horizontal position on the graph.
  • The y-coordinate describes the vertical position.
  • Intercepts are points where the line meets axes. Here, we found the x-intercept (-20, 0) and the y-intercept (0, -12).
Understanding these concepts helps us visualize and solve equations graphically. It bridges the gap between algebra and geometry.

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Most popular questions from this chapter

A sales person makes a base salary of \(\$ 2000\) per month. Once he reaches \(\$ 40,000\) in total sales, he earns an additional \(5 \%\) commission on the amount in sales over \(\$ 40,000 .\) Write a piecewise-defined function to model the sales person's total monthly salary \(S(x)\) (in \(\$)\) as a function of the amount in sales \(x\).

Evaluate the step function defined by \(f(x)=[x]\) for the given values of \(x\). $$ f(6) $$

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=5 x+9$$

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$(r+q)(x)$$

a. Given \(m(x)=4 x^{2}+2 x-3,\) find \(m(-x)\). b. Find \(-m(x)\). c. Is \(m(-x)=m(x)\) ? d. Is \(m(-x)=-m(x)\) ? e. Is this function even, odd, or neither?
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