Problem 65 - Algebra 1: Concepts and Skills

Question

Find three solutions of the equation. $$ y=-5 x+7 $$

Step-by-Step Solution

Answer

The three solutions of the equation \(y=-5x+7\) can be (0,7), (1,2), and (2,-3).

Step 1: Select a value for variable x

Pick any random number for x. For the beginning, let's start with x=0.

Step 2: Substitute x into Equation

Insert the chosen x value into the equation \(y=-5x+7\). So when x=0, substitute x=0 into the equation to get \(y=-5*(0)+7\).

Step 3: Solve for y

On simplification we get, y=7.

Step 4: Repeat step 1 to 3 for different x values

Repeat the same process by picking different x values. For instance, for x=1, substituting in the equation we get \(y=-5*(1)+7=-5+7=2\). Similarly, for x=2, substituting in the equation we get \(y=-5*(2)+7=-10+7=-3\) .

Key Concepts

Solutions of Equations

In mathematics, finding the solution of an equation involves determining the values of the variables that satisfy the equation. When dealing with linear equations, such as \( y = -5x + 7 \), the solutions are usually points on a straight line in the Cartesian plane. Here's how you can understand it:

• Each solution corresponds to a pair \((x, y)\) that makes the equation true.
• For instance, in the equation \( y = -5x + 7 \), if you plug in different values for \( x \), you will discover corresponding \( y \) values that solve the equation.
• A single linear equation with two variables, \( x \) and \( y \), generally has infinitely many solutions forming a line.

When you pick any number for \( x \), you can easily calculate \( y \) using the equation, leading to an ordered pair that is a part of the solution set.

Substitution Method

The substitution method is a powerful technique used to solve systems of equations, but it also applies to single equations with multiple variables when you're trying to find solutions. In our example of the equation \( y = -5x + 7 \):

• You begin by selecting a value for one of the variables. Typically, this is the \( x \) value since the equation is already solved for \( y \).
• Insert the chosen \( x \) value into the equation to find \( y \). For instance, substituting \( x = 0 \) gives \( y = 7 \).
• This process is repeated with different \( x \) values to generate other solutions.

Using substitution lets you methodically generate different solutions and understand the relationship between the variables in the equation. It makes solving straightforward linear equations much easier, especially when practicing with simple values like integers.

Solving for y

Solving for \( y \) in a linear equation means expressing \( y \) explicitly in terms of \( x \). Our equation, \( y = -5x + 7 \), is naturally arranged to solve for \( y \), which simplifies the process:

• With \( y \) on its own on one side of the equation, calculating its value for any \( x \) becomes simple arithmetic.
• By choosing different \( x \) values like \( 0, 1, \) or \( 2 \) and substituting them into the equation, you perform basic multiplication and addition to find corresponding \( y \) values.
• Each calculated \( y \) creates a valid solution pair along the line described by the equation.

This explicit form \( y = mx + b \), where \( m \) and \( b \) are constants, is called the slope-intercept form. It is particularly useful for graphing straight lines and understanding how changes in \( x \) affect \( y \). By consistently substituting and solving for \( y \), students grasp not only the mechanics but also the concepts involved in forming solutions.