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Chapter 2: Problem 34

Find the horizontal and vertical intercepts of each equation. $$ 7 x+2 y=56 $$

Short Answer

Expert verified
Horizontal intercept: (8, 0); Vertical intercept: (0, 28).

Step by step solution

01

Find the Horizontal Intercept

To find the horizontal intercept, set \( y = 0 \) in the equation \( 7x + 2y = 56 \). This gives us \( 7x + 2(0) = 56 \). Simplifying this, we have \( 7x = 56 \). Now, divide both sides by 7,\[ x = \frac{56}{7} = 8.\]Thus, the horizontal intercept is \( (8, 0) \).
02

Find the Vertical Intercept

To find the vertical intercept, set \( x = 0 \) in the equation \( 7x + 2y = 56 \). This gives us \( 7(0) + 2y = 56 \). Simplifying this, we have \( 2y = 56 \). Now, divide both sides by 2,\[ y = \frac{56}{2} = 28.\]Thus, the vertical intercept is \( (0, 28) \).

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Key Concepts

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

Horizontal Intercept
In mathematics, the horizontal intercept is a point where a graph intersects the x-axis. At this point, the value of the y-coordinate is zero. Calculating the horizontal intercept is quite straightforward. You need to set the y-value to zero in the equation and solve for x. Let's apply this principle to an example using the equation:
  • Set \( y = 0 \) in the equation \( 7x + 2y = 56 \) to get \( 7x + 2(0) = 56 \).
  • This simplifies to \( 7x = 56 \). When you solve it by dividing both sides by 7, you get \( x = 8 \).
  • Thus, the horizontal intercept is the point \( (8, 0) \).

Understanding the horizontal intercept helps you determine where the graph crosses the x-axis and is vital for analyzing linear equations.
Vertical Intercept
The vertical intercept is where a graph crosses the y-axis. At this position, the x-coordinate is zero. To find the vertical intercept, substitute zero for x in the equation and solve for y. Let's see how it's done:
  • Set \( x = 0 \) in the equation, resulting in \( 7(0) + 2y = 56 \).
  • This reduces to \( 2y = 56 \). Dividing both sides by 2 gives \( y = 28 \).
  • Therefore, the vertical intercept is \( (0, 28) \).

Knowing the vertical intercept is crucial in understanding the graph of an equation, as it indicates where the line moves through the y-axis.
Linear Equations
Linear equations are a straightforward type of algebraic equation that describes straight lines. They are represented in the form "ax + by = c," where a, b, and c are constants. With linear equations, both variables appear to the first power and have no products between them. In these equations:
  • The graph is always a straight line.
  • There are often two variables involved, typically x and y.
  • Their solutions include a set of points that lie on a straight line in a coordinate plane.

The simplicity of linear equations makes them an excellent starting point for learning more complex mathematical concepts like intercepts and slopes.
Solving Equations
Solving equations is the process of finding the value(s) of the variable(s) that make the equation true. This can include means such as simplifying expressions, isolating variables, and applying algebraic operations. Key steps often involve:
  • Simplifying each side of the equation by combining like terms and using the distributive property if necessary.
  • Isolating the variable by using inverse operations like addition/subtraction or multiplication/division to both sides of the equation.
  • Checking the solution by substituting it back into the original equation to verify it satisfies the equation.

Understanding how to solve equations is crucial for entire mathematical problem solving, whether finding intercepts, solving for unknowns, or modeling real-world situations.

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