Year 9 and 10 - Solving Linear Equations
MathsOnline・2 minutes read
The text provides solutions to five algebraic equations, demonstrating step-by-step methods for isolating variables and simplifying expressions. Each example illustrates different techniques such as rearranging terms, finding common denominators, and dividing to solve for the variable.
Insights
- The text illustrates the step-by-step process of solving various algebraic equations, emphasizing the importance of isolating the variable through addition, subtraction, multiplication, and division. Each example demonstrates a different method, showcasing that mathematical problems can often be approached in multiple ways, which can help build a deeper understanding of algebraic concepts.
- Additionally, the text highlights the necessity of using common denominators in equations involving fractions, as seen in the fourth and fifth examples. This technique not only simplifies the equations but also reinforces the idea that understanding the structure of the equation is crucial for finding the correct solution efficiently.
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Recent questions
What is algebra used for?
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is used to represent and solve problems involving relationships between quantities. Algebra allows for the formulation of equations that can model real-world situations, such as calculating distances, determining costs, or predicting outcomes. By using variables to stand in for unknown values, algebra provides a systematic way to analyze and solve problems, making it an essential tool in fields ranging from engineering and physics to economics and statistics.
How do you solve an equation?
Solving an equation involves finding the value of the variable that makes the equation true. This process typically includes isolating the variable on one side of the equation through a series of algebraic operations, such as addition, subtraction, multiplication, or division. For example, if you have an equation like \(5a - 2 = 28\), you would first add 2 to both sides to eliminate the constant term, resulting in \(5a = 30\). Then, you would divide both sides by 5 to solve for \(a\), yielding \(a = 6\). This method can be applied to various types of equations to find the unknowns.
What is a variable in math?
A variable in mathematics is a symbol, often represented by letters such as \(x\), \(y\), or \(a\), that stands for an unknown or changeable value. Variables are fundamental in algebra as they allow for the expression of general relationships and the formulation of equations. For instance, in the equation \(9x - 7 = 3x + 11\), \(x\) is the variable that we need to solve for. By manipulating the equation, we can determine the specific value of \(x\) that satisfies the equation, illustrating how variables are used to represent and solve mathematical problems.
What does it mean to isolate a variable?
Isolating a variable means rearranging an equation so that the variable is on one side of the equation by itself, making it easier to solve for that variable. This process often involves using inverse operations to eliminate other terms from the side of the equation where the variable is located. For example, in the equation \(6(x + 2) = 8(x + 1)\), you would first expand both sides and then rearrange the equation to isolate \(x\). This step is crucial in solving equations, as it allows you to directly determine the value of the variable in question.
What is a linear equation?
A linear equation is a type of equation that represents a straight line when graphed on a coordinate plane. It is characterized by the highest exponent of the variable being one, which means it can be expressed in the standard form \(ax + b = c\) or in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Linear equations can be solved using various methods, including graphing, substitution, or elimination, and they are fundamental in algebra as they model relationships with a constant rate of change, making them applicable in numerous real-world scenarios.
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Summary
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Solving Equations Step by Step
- The first example involves solving the equation \(5a - 2 = 28\). To isolate \(a\), add 2 to both sides, resulting in \(5a = 30\). Then, divide both sides by 5 to find \(a = 6\).
- In the second example, \(9x - 7 = 3x + 11\), rearrange the equation by moving \(-7\) to the right side (becoming \(+7\)) and \(3x\) to the left side (becoming \(-3x\)). This simplifies to \(6x = 18\). Dividing both sides by 6 gives \(x = 3\).
- The third example, \(6(x + 2) = 8(x + 1)\), requires expanding both sides: \(6x + 12 = 8x + 8\). Rearranging gives \(-2x = -4\). Dividing by \(-2\) results in \(x = 2\).
- The fourth example, \(\frac{x}{2} + \frac{x}{3} = 10\), involves finding a common denominator of 6. Multiplying through by 6 yields \(3x + 2x = 60\), simplifying to \(5x = 60\). Dividing by 5 results in \(x = 12\).
- In the fifth example, \(\frac{3}{5} - \frac{2}{2} = 5x\), the common denominator is 10. Multiplying through by 10 gives \(30 - 10 = 50x\), simplifying to \(20 = 50x\). Dividing by 50 results in \(x = \frac{22}{15}\) or \(1 \frac{7}{15}\).
