Understanding the Gradient and Y-Intercept of 2x + y - 3 = 0

To understand a linear equation better, we need to rearrange it into slope-intercept form.We start with the equation two x plus y minus three equals zero.Our first step is to move all x terms to the right side of the equation.When we subtract two x from both sides, the two x on the left becomes negative two x on the right.Next, we'll move the constant term. We add three to both sides to isolate y.Adding three to both sides cancels out the negative three on the left, giving us our final form.Now our equation is in slope-intercept form, which is y equals m x plus b.In this form, we can easily identify the coefficient of x and the constant term.Now that we have our equation in slope-intercept form, we can analyze its components.Now that we have our equation in slope-intercept form, let's focus on the coefficient of x.The coefficient negative two represents the gradient, or slope, of our line.A negative slope means that as we move right along the x-axis, the line goes down.Let's see exactly what negative two means. Starting from our y-intercept at three...When we move one unit right...The line goes down two units.This gives us our slope ratio: negative two over one, which simplifies to negative two.This pattern continues along the entire line. For every one unit we move right, we go down two units.The negative slope tells us that the line always goes down as we move right, making it a decreasing function.Now that we understand how the slope affects our line's direction and steepness, let's examine the y-intercept.Now that we understand the slope, let's examine the y-intercept in our equation y equals negative two x plus three.The constant term, positive three, represents the y-intercept. This is where our line crosses the y-axis.To find the y-intercept, we substitute x equals zero into our equation. This gives us the y-coordinate where the line crosses the y-axis.When we plot this point at zero comma three on our coordinate plane, we can see exactly where our line intersects the y-axis.Let's draw our line with slope negative two passing through this y-intercept point.Watch how any point on this line relates to our y-intercept. As we move along the line, the y-intercept remains fixed at three.Together with our slope of negative two, the y-intercept of three uniquely defines this line. No other line can have both these values.