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How to Solve Standard Form: Rules & Practice

Instructor: Amy Brozio-Andrews
Solving a standard form equation is a bit like cracking a walnut. The answers can be found once you crack into the shell by using the right formulas and rules. In this case, you need the slope and y-intercept formulas. Read on to learn how to use them.

Standard Equation, Slope, and Y-Intercept

Typically, when is solving a standard form equation, one is trying to find either its slope and/or its y-intercept.

The slope of an equation is the rate at which something changes. For example, let's say that you have a business that charges $0.50 per unit of goods sold. By knowing how many goods you sell, you can calculate with this value (the rate) how much money you would acquire.



The y-intercept of an equation is the starting point for when change is being measured. For example, using the same business example, you can calculate how much profit you make over a period of time by knowing how much money you started with before selling your goods. If you started with $15, you would deduct $15 from your total amount of money to calculate your amount of profit.



Let's proceed with looking at what a standard equation looks like. A standard form equation looks like this: Ax + By = C where A, B, and C represent numbers. For example, a standard equation with numbers looks like this: 5x - 3y = 8 (A = 5, B = -3, and C = 8). If you are asked to solve for either the slope or y intercept, you will need to utilize the following formulas:

Finding the Slope

To find the slope, divide the value of A by the value of B. The formula is: slope = A / B

The sign of the slope, if it is positive or negative, is derived from the following rules:

  • The slope is negative (-) if A and B have the same signs
  • The slope is positive (+) if A and B have different signs

Let's look at a sample problem : 7x - 3y = 4 (Ax + By = C)

The slope for this problem is found by dividing the values in front of x and y. In this case that is 7 and -3. Since 7 is positive while 3 is negative, the sign for the slope would be positive. Our slope for this equation = A / B = 7 / 3.

Here is another sample problem: -2x - 5y = 3 (Ax + By = C)

The slope for this problem is found by dividing the values in front of x and y. In this case that is -2 and -5. The sign for the slope would be negative since A and B are have the same sign, negative. Our slope for this equation = A / B = -2 / 5.

Now let's look at solving for the y-intercept.

Finding the Y-Intercept

To find the y-intercept, we divide the value of C by the value of B. This is the formula: y-intercept = C / B

The sign of the y-intercept (if it is positive or negative) is derived from the following rules:

  • The y-intercept is negative (-) if B and C have different signs
  • The y-intercept is positive (+) if B and C have the same signs

Let's look at an example: 10x -8y = 7 (Ax + By = C)

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