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Chapter 3: Problem 39

In the following exercises, find the intercepts for each equation.. $$ 3 x+y=6 $$

Short Answer

Expert verified
The x-intercept is (2, 0) and the y-intercept is (0, 6).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. \[ 3x + 0 = 6 \] This simplifies to: \[ 3x = 6 \] Divide both sides by 3: \[ x = 2 \] So, the x-intercept is (2, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. \[ 3(0) + y = 6 \] This simplifies to: \[ y = 6 \] So, the y-intercept is (0, 6).
03

Write the intercepts

Summarize the intercepts found from the previous steps. The intercepts are: 1. x-intercept: (2, 0) 2. y-intercept: (0, 6).

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Key Concepts

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

finding intercepts
Finding intercepts is an essential skill in algebra. Intercepts are points where a graph crosses the x-axis and y-axis. These points help understand the behavior of linear equations.
To find the x-intercept, set y to 0 in the equation and solve for x. This gives the point where the graph meets the x-axis.
For the y-intercept, set x to 0 and solve for y. This gives the point where the graph meets the y-axis.
Practically, intercepts are found using the steps:
  • Set y to 0 and solve for x to find the x-intercept.
  • Set x to 0 and solve for y to find the y-intercept.
In the given example, 3 x + y = 6:
  • x-intercept: Set y to 0; solve as follows, 3 x = 6; thus, x = 2, providing the point (2,0).
  • y-intercept: Set x to 0; solve as follows, y = 6, providing the point (0,6).
So, the intercepts are
x-intercept: (2, 0)
y-intercept: (0, 6).
solving linear equations
Solving linear equations is fundamental in algebra. A linear equation represents a straight line when graphed. Each solution to the equation is a point on this line.

When solving a linear equation like 3 x + y = 6, you aim to find the values of x and y that make the equation true.
There are several methods to solve a linear equation:
  • Solving for one variable in terms of the other, as shown when finding intercepts.
  • Graphical solution by plotting the line on a graph.
  • Elimination or substitution methods when dealing with systems of linear equations.
In our intercept finding, we effectively use substitution:
  • For x-intercept: Substitute y = 0 and solve.
  • For y-intercept: Substitute x = 0 and solve.
Understanding these methods enhances problem-solving skills and lays the groundwork for more complex algebraic challenges.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations. In the equation 3 x + y = 6, 3 x + y is the algebraic expression.

Understanding algebraic expressions is crucial because they represent relationships and patterns in math.
Some key points about algebraic expressions:
  • Terms: Parts of the expression separated by addition or subtraction, e.g., 3 x and y.
  • Coefficients: Numbers multiplying the variables, e.g., 3 in 3 x.
  • Constants: Standalone numbers, e.g., 6 in the equation.
Operations in algebraic expressions follow arithmetic rules but involve variables. To manage expressions efficiently, remember:
  • Combine like terms (same variable and exponent).
  • Use distributive property a(b + c) = ab + ac.
  • Factor expressions to simplify or solve equations.
Handling algebraic expressions aptly leads to solving equations, understanding functions, and tackling advanced algebra topics.

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