Linear Equations

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In the SAT Math section, linear equations are a common topic tested. Being able to solve linear equations efficiently and accurately is crucial for success. In this tutorial, we will cover the step-by-step process of solving linear equations, along with some helpful tips and strategies to tackle SAT linear equation problems.

Solving Linear Equations

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Let's use the below example to understand the every step as we learn them.

Solve the equation: $2x + 5 = 11$

1. Understand the Basics

a. A linear equation is an equation of a straight line and can be written in the form: ax + b = c, where a, b and c are constants and x is the variable.

b. The goal is to find the value of x that satisfies the equation.

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In our example, a is 2, b is 5 and c is 11.

2. Simplify the Equation

a. If the equation has any parentheses or fractions, begin by simplifying and clearing them.

b. Apply the distributive property to eliminate parentheses and find a common denominator to simplify fractions.

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The given equation is already simplified, so we can move on to the next step.

3. Isolate the Variable

a. Start by getting rid of any constant terms on the same side as the variable.

b. Use inverse operations to isolate the variable term (x) on one side of the equation. Remember that whatever operation you perform on one side, you must do the same to the other side to maintain equality.

c. Simplify both sides of the equation as much as possible.

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We want to isolate the 'x' term on one side of the equation. In this case, we'll move the constant term to the other side. To do this, subtract 5 from both sides of the equation:

$2x + 5 - 5 = 11 - 5$

$2x = 6$

4. Solve for the Variable

a. Once the variable is isolated on one side of the equation, apply appropriate operations to find its value.

b. If there are any like terms that can be combined, simplify further.

c. If necessary, divide both sides of the equation by a coefficient to obtain the value of 'x'.

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To find the value of 'x', we need to isolate it on one side of the equation. Since '2' is multiplied by 'x', we'll divide both sides by '2':

$\frac{(2x)}{2} = \frac{6}{2}$

$x = 3$

5. Check Your Solution

a. Substitute the found value of 'x' back into the original equation and ensure that it satisfies the equation.

b. If the substituted value satisfies the equation, it is the correct solution. If not, recheck your steps or calculations.

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To ensure the solution is correct, substitute the found value of 'x' back into the original equation and verify that it satisfies the equation:

$2(3) + 5 = 11$

$6 + 5 = 11$

$11 = 11$

The equation is true, so the solution $x = 3$ is correct.

Handling Equations with No Solution or Infinite Solutions

a. Some linear equations may not have a solution or may have infinitely many solutions.

b. If, after simplifying and solving the equation, you end up with a statement that is always false (e.g., 2 = 4), it means the equation has no solution.

c. If you get an identity (e.g., 3 = 3), it means the equation has infinitely many solutions, as any value of 'x' will satisfy the equation.

Tips for SAT Linear Equation Problems

a. Practice solving linear equations regularly to build speed and accuracy.

b. Be cautious with negative signs, especially when multiplying or dividing by negative numbers.

c. Keep an eye out for common mistakes such as errors in arithmetic or sign errors.

d. Pay attention to the wording of the problem to identify relevant information.

e. Use your scratch paper to show all steps and calculations, minimizing errors.

More Examples

Example 1

Solve the equation: $3x - 2 = 4x + 1$

Step 1: Understand the Basics

The equation is in the form ax + b = c, where a is the coefficient of x, b is a constant term, and c is the value on the other side of the equation. Our goal is to find the value of x that satisfies the equation.

Step 2: Simplify the Equation

Start by moving all terms containing x to one side and all constant terms to the other side:

$3x - 2 = 4x + 1$

To do this, subtract 4x from both sides of the equation:

$3x - 4x - 2 = 4x - 4x + 1$

Simplify further:

$-x - 2 = 1$

Step 3: Isolate the Variable

To isolate the variable x, we'll move the constant term to the other side. Add 2 to both sides of the equation:

$-x - 2 + 2 = 1 + 2$

Simplify further:

$-x = 3$

Step 4: Solve for the Variable

To solve for x, multiply both sides of the equation by -1 to get rid of the negative sign:

$(-1)(-x) = (-1)(3)$

Simplify:

$x = -3$

Step 5: Check Your Solution

To verify the result, substitute the found value of x back into the original equation and check if the equation holds true:

$3(-3) - 2 = 4(-3) + 1$

Simplify:

$-9 - 2 = -12 + 1$

$-11 = -11$

The equation holds true, which confirms that x = -3 is the correct solution.

Example 2

Solve the equation: $3(2x - 5) + 4(3 - x) = 2(x + 1) - 5$

Step 1: Understand the Basics

The equation is in the form ax + b = c, where 'a' is the coefficient of 'x', 'b' is a constant term, and 'c' is the value on the other side of the equation. Our goal is to find the value of 'x' that satisfies the equation.

Step 2: Simplify the Equation

Start by applying the distributive property to simplify each side of the equation:

$3(2x - 5) + 4(3 - x) = 2(x + 1) - 5$

$6x - 15 + 12 - 4x = 2x + 2 - 5$

Combine like terms on each side of the equation:

$6x - 4x - 15 + 12 = 2x - 3$

$2x - 3 = 2x - 3$

Step 3: Isolate the Variable

In this equation, both sides are already simplified and the variable x is already isolated. We can move on to the next step.

Step 4: Solve for the Variable

Since both sides of the equation are identical, we can conclude that this equation has infinitely many solutions. Any value of 'x' will satisfy the equation.

Step 5: Check Your Solution

To verify the result, we can substitute any value for 'x' and check if the equation holds true. Let's choose x = 0 for this example:

$3(2(0) - 5) + 4(3 - 0) = 2(0 + 1) - 5$

$3(-5) + 4(3) = 2(1) - 5$

$-15 + 12 = 2 - 5$

$-3 = -3$

The equation is true, which confirms that x = 0 is a valid solution.

Conclusion

Mastering the skill of solving linear equations is essential for success in the SAT Math section. By following the step-by-step process outlined in this tutorial and practicing regularly, you'll become more confident in handling SAT linear equation problems. Remember to pay attention to the details and double-check your solutions to ensure accuracy. Good luck!

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