Problem 30 - Algebra 1

Question

Write the equation in standard form with integer coefficients. $$y=-\frac{3}{4} x+\frac{5}{4}$$

Step-by-Step Solution

Answer

The equation in standard form is \(3x + 4y = 5\).

Step 1: Eliminate the Fraction

Multiply through by 4, which is the common denominator of the fractions, to eliminate the fractions. This gives \(4y = -3x + 5\).

Step 2: Rearrange terms

Rearrange the equation so that the x and y terms are all on the left side. This gives \(3x + 4y -5 = 0\).

Step 3: Final Equation

Check if the coefficient of x is positive. If not, multiply through by -1 but in this case, coefficient of x is positive. Therefore, the equation in standard form is \(3x + 4y = 5\).

Key Concepts

Eliminating Fractions

When dealing with equations in algebra, one often encounters fractions. Fractions can make equations look complicated and more difficult to solve. A method to simplify the equation and make it easier to work with is to eliminate fractions. To do so, you need to find a common denominator which is a multiple of all the denominators in the equation.

Take the equation from our exercise, \(y = -\frac{3}{4}x + \frac{5}{4}\). The common denominator here is 4. Multiplying each term by this common denominator clears the fractions, resulting in \(4y = -3x + 5\). This makes the subsequent steps to solve the equation more straightforward since you're now working with whole numbers.

It's crucial to apply the multiplication across the entire equation to maintain balance. Each term, on both sides of the equation, must be multiplied by the common denominator to ensure that the equation remains equivalent to the original.

Rearranging Equations

Rearranging an equation is a fundamental algebraic skill. To convert an equation into its standard form, which typically looks like \(Ax + By = C\), where A, B, and C are integers, you often need to rearrange the terms. After eliminating fractions in our example, we obtained \(4y = -3x + 5\). From there, we need to rearrange the terms to get the x and y terms on one side and the constant on the other side.

To achieve this, we add 3x to both sides, which yields \(3x + 4y = 5\). This step is crucial because standard form requires that all variable terms are on one side of the equation, creating a clearer picture of the relationship between x and y. Remember, whatever operation you perform on one side of the equation, you must do the same to the other side to maintain the equation's balance.

Integer Coefficients

Integer coefficients in equations make them easier to interpret and often are required for standard form equations. Standards form is usually written as \(Ax + By = C\), where A, B, and C are integers without any common factors other than 1. It's preferable that A is positive. If you reach a point where your coefficient is negative, as in \(-3x + 4y = -5\), you can multiply the entire equation by -1 to make the coefficient positive.

In our exercise, after rearranging the terms, we obtained \(3x + 4y = 5\). Here, the coefficients 3, 4, and 5 are already integers, and 3 is positive. This equation is thus in the standard form with integer coefficients. It is particularly helpful when graphing lines, solving systems of equations, and performing other algebraic operations that require a clear and simple equation form.