Math Problem Statement

give me the standard linear form and identity A,B,C.

Write the following in a standard form and identify A, B, C.

Solution

To convert the equations from your image into their standard form and identify the coefficients AA, BB, and CC, let's analyze each equation.

General Standard Form for Linear Equations:

The general form of a linear equation in two variables is: Ax+By=CAx + By = C Where:

  • AA, BB, and CC are constants.
  • xx and yy are variables.

Equation 1:

The given equation is: 1.5y=9x91.5y = 9x - 9 Rearrange this to fit the standard form: 9x+1.5y=9-9x + 1.5y = -9 Multiply the entire equation by 2 to remove the decimal: 18x+3y=18-18x + 3y = -18 Now, the equation is in standard form. So:

  • A=18A = -18
  • B=3B = 3
  • C=18C = -18

Equation 2:

The given equation is: 212y+5=6y2 \cdot \frac{1}{2}y + 5 = 6y Simplify the left side: y+5=6yy + 5 = 6y Rearrange to bring terms involving yy on one side: y6y=5y - 6y = -5 5y=5-5y = -5 So, divide by 5-5: y=1y = 1 This equation is not linear in terms of both xx and yy, but rather expresses yy as a constant.

Equation 3:

The given equation is: 2x+4=6y2x + 4 = 6y Rearrange to fit the standard form: 2x6y=42x - 6y = -4 Thus, in standard form:

  • A=2A = 2
  • B=6B = -6
  • C=4C = -4

Equation 4:

The given equation is: 2x=4y82x = 4y - 8 Rearrange to fit the standard form: 2x4y=82x - 4y = -8 Thus, in standard form:

  • A=2A = 2
  • B=4B = -4
  • C=8C = -8

Equation 5:

The given equation is: 3y6=4x3y - 6 = 4x Rearrange to fit the standard form: 4x3y=64x - 3y = -6 Thus, in standard form:

  • A=4A = 4
  • B=3B = -3
  • C=6C = -6

Summary of Coefficients AA, BB, and CC:

  1. Equation 1: A=18A = -18, B=3B = 3, C=18C = -18
  2. Equation 2: y=1y = 1 (Not in the general linear form)
  3. Equation 3: A=2A = 2, B=6B = -6, C=4C = -4
  4. Equation 4: A=2A = 2, B=4B = -4, C=8C = -8
  5. Equation 5: A=4A = 4, B=3B = -3, C=6C = -6

Would you like to ask more about any of these solutions or require additional explanations?

Related Questions:

  1. How can you solve a system of linear equations with these forms?
  2. What are the key properties of linear equations in two variables?
  3. How can graphing help in visualizing solutions of these equations?
  4. What is the difference between consistent, inconsistent, and dependent systems?
  5. How do changes in coefficients AA, BB, or CC affect the line's slope and intercepts?

Tip:

When rearranging equations to the standard form, always ensure the coefficient of xx is on the left, and watch for sign changes when moving terms across the equal sign.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Standard Form of Equations

Formulas

Ax + By = C (Standard Form)

Theorems

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Suitable Grade Level

Grades 8-10