Problem 11
Question
Solve and check linear equation. \(3(x-4)-4(x-3)=x+3-(x-2)\)
Step-by-Step Solution
Verified Answer
The solution for the linear equation is \(x=-2.5\).
1Step 1: Distribution
First distribute on both sides of the equation. That is, multiply each term inside the brackets by the number outside it. This gives \(3x-12-4x+12=x+3-x+2\).
2Step 2: Simplify the expression
Next, simplify by adding or subtracting the like terms on both sides, which gives \(-x=x+5\).
3Step 3: Isolate variable x
After that, combine like terms on both sides of the equation, moving all x terms to one side, and this gives \(2x=-5\).
4Step 4: Solving for x
Lastly, solve for x by dividing by 2 on both sides of the equation, to have \(x=-2.5\).
Key Concepts
Distributive PropertyCombining Like TermsIsolating Variables
Distributive Property
The distributive property is a powerful tool in algebra that allows us to multiply a single term by each term within a parenthesis. In other words, it distributes the multiplication over addition or subtraction. For instance, if you have a problem like \(3(x-4)\), you would apply the distributive property to multiply 3 by \(x\) and \( -4 \) separately. This process turns \(3(x-4)\) into \(3x - 12\), as 3 is multiplied by \(x\) and \( -4\) consecutively.
Applying the distributive property correctly is crucial for simplifying and solving equations. As in the given exercise, we distribute on both sides to eliminate the brackets and simplify the equation step by step. The action of distribution transforms the equation into one that is easier to manage and sets the stage for combining like terms and further manipulation to solve for the variable.
Applying the distributive property correctly is crucial for simplifying and solving equations. As in the given exercise, we distribute on both sides to eliminate the brackets and simplify the equation step by step. The action of distribution transforms the equation into one that is easier to manage and sets the stage for combining like terms and further manipulation to solve for the variable.
Combining Like Terms
Combining like terms is the process of simplifying algebraic expressions by adding or subtracting terms that have the same variables raised to the same power. Essential for simplification, this step reduces the complexity of the problem by consolidating similar elements.
In the context of the provided exercise, after distributing, we have terms \(3x\), \( -4x\), and \(x\), which are all like terms since they contain the variable \(x\) to the same power. Hence, they can be combined. Likewise, constants without variables (like \( -12\), \(12\), and the numbers \( 3\) and \(2\) on the other side of the equation) are also like terms and can be combined. The equation simplifies further by adding and subtracting these like terms, resulting in a more straightforward equation, \( -x = x + 5\), which then leads to isolating the variable.
In the context of the provided exercise, after distributing, we have terms \(3x\), \( -4x\), and \(x\), which are all like terms since they contain the variable \(x\) to the same power. Hence, they can be combined. Likewise, constants without variables (like \( -12\), \(12\), and the numbers \( 3\) and \(2\) on the other side of the equation) are also like terms and can be combined. The equation simplifies further by adding and subtracting these like terms, resulting in a more straightforward equation, \( -x = x + 5\), which then leads to isolating the variable.
Isolating Variables
Isolating the variable is the final crucial step in solving linear equations. It involves moving all instances of the variable to one side of the equation and all constants to the other side. This is typically achieved through addition, subtraction, division, or multiplication, allowing us to 'isolate' the variable, a clear path towards finding its value.
In the exercise at hand, after combining like terms, the process entails moving the terms containing \(x\) to one side and the constants to the opposite side to get the variable by itself. As shown, we go from \( -x = x + 5\) to \(2x = -5\) by moving all the \(x\) terms to one side and combining them. Finally, we divide both sides by the coefficient of \(x\), which is 2 in this case, yielding the solution \( x = -2.5\). This method helps to clearly define the solution, ensuring that the variable of interest is highlighted and its value is straightforward to deduce.
In the exercise at hand, after combining like terms, the process entails moving the terms containing \(x\) to one side and the constants to the opposite side to get the variable by itself. As shown, we go from \( -x = x + 5\) to \(2x = -5\) by moving all the \(x\) terms to one side and combining them. Finally, we divide both sides by the coefficient of \(x\), which is 2 in this case, yielding the solution \( x = -2.5\). This method helps to clearly define the solution, ensuring that the variable of interest is highlighted and its value is straightforward to deduce.
Other exercises in this chapter
Problem 11
Express each interval in set-builder notation and graph the interval on a number line. $$(-\infty, 3)$$
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Check all proposed solutions. \(\sqrt{3 x+18}=x\)
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In Exercises \(1-12\), plot the given point in a rectangular coordinate system. $$ \left(\frac{2}{2},-\frac{3}{2}\right) $$
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In Exercises \(9-20,\) find each product and write the result in standard form. $$(-4-8 i)(3+i)$$
View solution