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Find the slope of the line −2𝑥 + 3𝑦 − 2 = 0 and the 𝑦-intercept of this line.

Find the slope of the line negative two 𝑥 plus three 𝑦 minus two equals zero and the 𝑦-intercept of this line.

The most efficient way to find both the slope and the 𝑦-intercept of a straight line is to rearrange its equation into the slope–intercept form. That’s 𝑦 equals 𝑚𝑥 plus 𝑏, where the value of 𝑚, that’s the coefficient of 𝑥, gives the slope of the line and the value of 𝑏, the constant term, gives its 𝑦-intercept. Let’s take the equation we’ve been given then and rearrange it to make 𝑦 the subject. We’ll begin by adding both two 𝑥 and two to each side of this equation as this will eliminate the negative two 𝑥 and the negative two on the left-hand side. We could do this in two stages if we prefer. This gives three 𝑦 is equal to two 𝑥 plus two.

The next step is to divide both sides of the equation by three so that the coefficient of 𝑦 is one. This gives 𝑦 equals two-thirds 𝑥 plus two-thirds. And the equation of this line is now in its slope–intercept form. The slope of this line is the coefficient of 𝑥, which we see is equal to two-thirds. And the 𝑦-intercept is the constant term, which we can see is also equal to two-thirds. So by rearranging the equation of this line into the slope–intercept form 𝑦 equals 𝑚𝑥 plus 𝑏, we found that both the slope and the 𝑦-intercept of the line negative two 𝑥 plus three 𝑦 minus two equals zero are two-thirds.

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