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 polynomials, 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\(. Then, combine the constants on the left side to simplify it to \)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 means 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\( is multiplied by 2, divide both sides by 2 to find \)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 is confirmed as correct.

Mastering this methodical approach to solving linear equations enhances problem-solving skills and builds a strong foundation for algebra. Consistent practice with these steps—simplifying expressions, applying properties of equality, isolating variables, and verifying solutions—ensures confidence and accuracy in solving any linear equation encountered.

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 to isolate terms. Finally, apply multiplication or division properties to solve for the variable by eliminating coefficients. For instance, divide both sides by the coefficient of $x$. Always check your solution by substituting the value back into the original equation to verify it produces a true statement. This process ensures accuracy and reinforces understanding of algebraic principles.

How do you use the properties of equality to isolate the variable in a linear equation?

The properties of equality allow you to manipulate equations while maintaining balance. To isolate the variable, first use the addition or subtraction property to move all variable terms to one side and constants to the other. For example, if your equation is $\mathrm{3x}-4=x+8$, subtract $x$ from both sides to combine variable terms, and add $4$ to both sides to combine constants. Then, use the multiplication or division property to eliminate coefficients attached to the variable. If $2x\; =\; 12$, divide both sides by 2 to get $x\; =\; 6$. These steps ensure the variable stands alone, making it easier to find its value.

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

Checking the solution of a linear equation is crucial because it confirms the accuracy of your answer. After solving for the variable, substitute the value back into the original equation. If both sides of the equation are equal, your solution is correct. For example, if you found $x\; =\; 6$ for the equation $3(x\; -\; 2)\; +\; 2\; =\; x\; +\; 8$, substitute 6 for $x$ to verify: $3(6\; -\; 2)\; +\; 2\; =\; 6\; +\; 8$ simplifies to $14\; =\; 14$, a true statement. This step helps catch mistakes made during simplification or manipulation and reinforces understanding of the equation's structure.

How do you simplify expressions when solving linear equations?

Simplifying expressions is the first step in solving linear equations. Begin by distributing any coefficients across parentheses. For example, in $3(x\; -\; 2)\; +\; 2\; =\; x\; +\; 8$, distribute 3 to get $3x\; -\; 6\; +\; 2\; =\; x\; +\; 8$. Next, combine like terms, especially constants. Here, $-6\; +\; 2$ simplifies to $-4$, so the equation becomes $3x\; -\; 4\; =\; x\; +\; 8$. Simplifying makes the equation easier to work with and prepares it for applying properties of equality to isolate the variable.

What is the role of combining like terms in solving linear equations?

Combining like terms simplifies an equation by merging terms that have the same variable or are constants. This reduces complexity and makes it easier to isolate the variable. For example, in the expression $3x\; -\; 6\; +\; 2$, the constants $-6$ and $2$ are like terms and combine to $-4$, resulting in $3x\; -\; 4$. Without combining like terms, equations become cluttered and harder to solve. This step is essential before applying properties of equality to isolate the variable and find the solution.