Problem 17 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form with integer coefficients. \(y=-9+4 x\)

Step-by-Step Solution

Answer

The standard form of the equation with integer coefficients is \(4x - y = 9\).

Step 1: Transposition of terms

To write the equation in standard form, need to rearrange terms so that the equation looks like this: \(Ax + By = C\). Start by subtracting 4x from both sides of the equation to move the x term to the left hand side: \(-4x + y = -9\).

Step 2: Make Sure Coefficient of x is Positive

It’s conventional to have the coefficient of the x term be a positive number. Therefore, need to multiply every term by -1, which would not change the equation. So the equation will be \(4x - y = 9\).

Key Concepts

Transposition of Terms

When we talk about transposing terms in algebra, we mean the process of moving a term from one side of an equation to the other, which helps in the simplification or rearrangement of the equation. It's a bit like sorting out pieces of a puzzle to get a clearer picture. Therefore, rearranging the original equation \(y = -9 + 4x\) involves shifting terms across the equality sign to establish the standard form, \(Ax + By = C\), where A, B, and C are integers, and A should be nonnegative.

For transposition, if we want to move a term to the other side, we do the opposite operation. In the example, the term \(4x\) is on the right-hand side of the equation with the operation 'addition.' To transpose it to the left-hand side, subtraction is used, resulting in \(-4x + y = -9\). Think of this step as balancing scales: what you do to one side, you must equally do to the other to maintain balance.

Integer Coefficients

The concept of integer coefficients pertains to the values that are placed in front of the variables in an equation. These coefficients are multiples that tell us how many times a variable (or unit) is counted. For instance, in the term \(4x\), the integer coefficient is 4, indicating we have '4 times x.' What's essential here is the requirement for these coefficients to be integers; in other words, whole numbers that can be positive, negative, or zero, without fractions or decimals -- making the equations more standardized and easier to compare or manipulate.

In our exercise, after transposing terms, both sides of the equation \(-4x + y = -9\) already have integer coefficients. It's crucial for equations in standard form because it ensures that anyone working with the equation is dealing with whole numbers, which simplifies further mathematical operations and the interpretation of results.

Positive Coefficient

Ensuring a positive coefficient is generally a standard mathematical convention when it comes to writing linear equations. A positive coefficient for the variable makes it easier to interpret and avoids unnecessary confusion. In the equation \(4x - y = 9\), the coefficient of \(x\) is positive, aligning with this convention.

Multiplying by -1

If you have a negative coefficient for your leading variable and you want to make it positive (like in the equation \(-4x + y = -9\)), you can multiply the entire equation by -1 to reverse the sign (\(-1 \times -4x = 4x\), \(-1 \times y = -y\), and \(-1 \times -9 = 9\)). This operation changes nothing about the equation's meaning but refines its form. So, when we take the initial equation, \(-4x + y = -9\), and multiply all terms by -1, we're adhering to convention by transforming the coefficient of the variable \(x\) into a positive one, resulting in the neatly packed standard-form equation: \(4x - y = 9\).