Solving Linear Equations: Videos & Practice Problems

Solving linear equations involves simplifying expressions by distributing and combining like terms, then using properties of equality to isolate the variable. Start by distributing coefficients across parentheses and combining constants. Next, apply addition or subtraction properties to gather variable terms on one side and constants on the other. Finally, use multiplication or division properties to solve for the variable. Checking the solution by substituting back into the original equation ensures accuracy. This process reinforces understanding of terms, coefficients, and the structure of polynomial expressions, essential for mastering algebraic problem-solving.

Strategy for Solving Linear Equations

Solving linear equations involves a systematic approach that combines simplification and the application of properties of equality to isolate the variable. The process begins by simplifying both sides of the equation, which includes distributing any coefficients across parentheses and combining like terms. For example, when given an equation such as \$3(x - 2) + 2 = x + 8\(, start by distributing the 3 to both \)x\( and \)-2\(, resulting in \)3x - 6 + 2 = x + 8\(. Then, combine the constants on the left side to simplify it further to \)3x - 4 = x + 8\(.

Next, use the addition and subtraction properties of equality to gather all variable terms on one side and constants on the other. This involves subtracting \)x\( from both sides to get \)3x - x - 4 = 8\(, which simplifies to \)2x - 4 = 8\(. Then, add 4 to both sides to isolate the term with the variable, yielding \)2x = 12\(.

To fully isolate the variable, apply the multiplication or division properties of equality. Since \)x\( is multiplied by 2, divide both sides by 2 to solve for \)x\(: \)x = \frac{12}{2} = 6\(. This step ensures the variable stands alone on one side of the equation, providing the solution.

Finally, always verify the solution by substituting the value back into the original equation. Plugging \)x = 6\( into \)3(x - 2) + 2 = x + 8\( gives \)3(6 - 2) + 2 = 6 + 8\(, which simplifies to \)3 \times 4 + 2 = 14\( and \)6 + 8 = 14\(. Since both sides equal 14, the solution \)x = 6$ is confirmed correct.

This methodical approach to solving linear equations—simplifying expressions, using addition and subtraction properties to group like terms, applying multiplication or division to isolate the variable, and verifying the solution—builds a strong foundation for tackling a wide range of algebraic problems with confidence and accuracy.

Solve the given linear equation. Check your solution.
$2(x+3)=14$

• A. $x=11$
• B. $x=4$
• C. $x=10$
• D. $x=5.5$

Answer: (B)

Solve the given linear equation. Check your solution.
$-5-y=3(y+9)$

• A. $y=-8$
• B. $y=8$
• C. $y=-3$
• D. $y=3$

Answer: (A)

Solve the given linear equation. Check your solution.
$4(x+1)-3(x-2)=2x+5$

• A. $x=3$
• B. $x=0$
• C. $x=-5$
• D. $x=5$

Answer: (D)

Here’s what students ask on this topic:

What are the step-by-step methods to solve a linear equation?

To solve a linear equation, start by simplifying both sides. This involves distributing any coefficients across parentheses and combining like terms, especially constants. Next, use the addition or subtraction properties of equality to gather all variable terms on one side and constants on the other. For example, if you have $\mathrm{3x}-4=x+8$, subtract $x$ from both sides and add $4$ to both sides. Finally, apply multiplication or division properties to isolate the variable. If the variable is multiplied by a number, divide both sides by that number. After finding the solution, always check it by substituting the value back into the original equation to ensure it produces a true statement. This systematic approach helps solve any linear equation efficiently.

How do you use the distributive property when solving linear equations?

The distributive property is used to eliminate parentheses in linear equations. It involves multiplying a single term outside the parentheses by each term inside. For example, in the equation $3(x-2)\; +2=x+8$, distribute the 3 to both $x$ and $-2$, resulting in $3x\; -\; 6\; +\; 2\; =\; x\; +\; 8$. After distribution, combine like terms such as constants $-6\; +\; 2\; =\; -4$. This simplification prepares the equation for the next steps of isolating the variable. Proper use of the distributive property is essential for correctly simplifying and solving linear equations.

What properties of equality are used to isolate the variable in linear equations?

To isolate the variable in a linear equation, we use three main properties of equality: addition, subtraction, and multiplication/division. First, use the addition or subtraction property to move all variable terms to one side and constants to the other. For example, subtract $x$ from both sides to combine variable terms. Then, use addition or subtraction to move constants accordingly. Finally, apply multiplication or division to eliminate coefficients attached to the variable. For instance, if the variable is multiplied by 2, divide both sides by 2 to solve for the variable. These properties maintain the equality while simplifying the equation to find the variable's value.

Why is it important to check the solution of a linear equation?

Checking the solution of a linear equation is crucial to verify its accuracy. After solving for the variable, substitute the solution back into the original equation. This ensures that both sides of the equation are equal, confirming the solution is correct. For example, if you find $x=\; 6$, plug 6 into the original equation and simplify both sides. If both sides are equal, the solution is valid. This step helps catch any errors made during simplification or manipulation and reinforces understanding of the problem-solving process.

How do you combine like terms when solving linear equations?

Combining like terms means adding or subtracting terms that have the same variable raised to the same power or are constants. For example, in the expression $3x\; -\; 6\; +\; 2$, the constants $-6$ and $2$ are like terms and can be combined to get $-4$. Variable terms like $3x$ and $x$ can also be combined by adding their coefficients, such as $3x\; -\; x\; =\; 2x$. Combining like terms simplifies the equation, making it easier to isolate the variable and solve the equation efficiently.