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Chapter 2: Problem 1

Find the values of \(y=2-3 x\) for: (i) \(x=0\) (ii) \(x=2\) (iii) \(x=-3\) (iv) \(x=2 / 3\)

Short Answer

Expert verified
The values of \(y\) are 2, -4, 11, and 0 for \(x=0, 2, -3,\) and \(\frac{2}{3}\) respectively.

Step by step solution

01

Substitute Value in Equation (i)

We start by substituting the given value of \(x=0\) into the equation. Replace \(x\) in \(y=2-3x\) with 0. So,\[y = 2 - 3(0) = 2 - 0 = 2.\]
02

Substitute Value in Equation (ii)

Next, replace \(x\) with 2 in the equation \(y=2-3x\). Substitute as follows:\[y = 2 - 3(2) = 2 - 6 = -4.\]
03

Substitute Value in Equation (iii)

Now, substitute \(x = -3\) into the equation \(y=2-3x\). Perform the calculation:\[y = 2 - 3(-3) = 2 + 9 = 11.\]
04

Substitute Value in Equation (iv)

Finally, substitute \(x = \frac{2}{3}\) into the equation \(y=2-3x\). Solve it as follows:\[y = 2 - 3\left(\frac{2}{3}\right) = 2 - 2 = 0.\]

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Key Concepts

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

Algebra
Algebra is a branch of mathematics that uses symbols to represent numbers in equations and formulas. One of the foundational concepts in algebra involves manipulating these symbols to solve for unknown variables.
In solving linear equations like \( y = 2 - 3x \), we use simple operations such as addition, subtraction, multiplication, and division to find solutions. Linear equations are called so because they graph as straight lines on a coordinate plane.
To work with algebra effectively, you often need to:
  • Understand how to substitute numbers for variables.
  • Perform order of operations correctly (often remembered by the acronym PEMDAS - Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
  • Be comfortable with both positive and negative numbers and their interactions with each other.
Each step in algebra builds on the previous, leading students to more complex problem-solving techniques.
Substitution Method
The substitution method is a technique used to solve equations, especially in algebra. It involves replacing a variable in an equation with a given value. This makes it easier to solve for the remaining variable.
When applying the substitution method, keep these strategies in mind:
  • Identify the variable you need to substitute. For example, in the exercise \( y = 2 - 3x \), \( x \) is the variable we replaced with different values.
  • Carefully substitute the value into the equation and ensure all arithmetic operations are executed following the order of operations.
  • Check your work by substituting back to ensure it satisfies the original equation.
This method is straightforward, making it a preferred choice when simple values are provided, reducing complex problems into simpler arithmetic tasks.
Equations of Lines
Equations like \( y = 2 - 3x \) represent lines on a graph. These are called linear equations. The form \( y = mx + b \), known as the slope-intercept form, is a common method for expressing linear equations.
In an equation of a line, you'll find:
  • \( m \), the slope, which shows the rate of change or how steep the line is. In \( y = 2 - 3x \), the slope is -3, meaning the line falls as it moves from left to right.
  • \( b \), the y-intercept, which is where the line crosses the y-axis. In \( y = 2 - 3x \), the y-intercept is 2, indicating the line intersects the y-axis at point (0,2).
Graphing these equations helps visualize the relationship between variables. It highlights how changing \( x \) affects \( y \), showing the linear connection between the variables.

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Most popular questions from this chapter

Simplify if possible:\(\frac{x^{2}-9}{x^{2}+5 x+6}\)

Factorize:\(4 x^{2}-9\)

Find the roots and sketch the graphs of the quadratic functions:\(-3 x^{2}+3 x-1\)

Make a table of \((x, y)\) values and sketch a fully labelled graph of the quadratic:\(y=x^{2}-4 x\)

Solve the simultaneous equations:\(3 x-2 y=1,2 x+3 y=2\)
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