A linear equation is a mathematical equation in which the variables are raised to the power of one. These equations represent a straight line when graphed, and they are characterized by a constant rate of change between the variables.
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Linear equations can be solved using the properties of equality, such as the Division and Multiplication Properties.
When solving linear equations with integers, the goal is to isolate the variable by performing the same operations on both sides of the equation.
Solving linear equations with decimals involves the same principles as solving equations with integers, but the decimal places must be carefully managed.
The Division and Multiplication Properties of Equality are used to solve linear equations with fraction or decimal coefficients by eliminating the fractions or decimals.
The solution to a linear equation represents the value of the variable that makes the equation true.
Review Questions
Explain how the properties of equality can be used to solve linear equations with integers.
When solving linear equations with integers, the goal is to isolate the variable by performing the same operations on both sides of the equation. This can be done using the properties of equality, such as the Addition Property (adding the same number to both sides) and the Multiplication Property (multiplying both sides by the same non-zero number). By applying these properties, you can eliminate the terms with the variable and find the value of the variable that satisfies the equation.
Describe the process of solving linear equations with decimal coefficients.
Solving linear equations with decimals involves the same principles as solving equations with integers, but the decimal places must be carefully managed. The goal is still to isolate the variable by performing the same operations on both sides of the equation. However, when dealing with decimal coefficients, you must ensure that the decimal places are aligned properly and that any operations, such as division or multiplication, are carried out correctly to maintain the integrity of the equation.
Analyze the role of the Division and Multiplication Properties of Equality in solving linear equations with fraction or decimal coefficients.
The Division and Multiplication Properties of Equality are essential in solving linear equations with fraction or decimal coefficients. By applying these properties, you can eliminate the fractions or decimals and simplify the equation to a form that is easier to solve. The Division Property allows you to divide both sides of the equation by the same non-zero number, while the Multiplication Property allows you to multiply both sides by the same non-zero number. This process helps to isolate the variable and find the solution to the linear equation.