We have a linear equation with fractions. One third times x minus three, minus one fifth times x minus five, equals three fourths times three x minus four, plus five. Our goal is to find the value of x by eliminating the fractions first.
Step one is to clear the fractions. The denominators are three, five, and four. The least common multiple of these numbers is sixty. We multiply the entire equation by sixty. This gives us twenty times x minus three, minus twelve times x minus five, equals forty five times three x minus four, plus three hundred.
Step two is to distribute. On the left side, twenty times x minus three gives twenty x minus sixty. Minus twelve times x minus five gives minus twelve x plus sixty. On the right side, forty five times three x minus four gives one hundred thirty five x minus one hundred eighty, plus three hundred. Step three is to combine like terms. Twenty x minus twelve x equals eight x. Negative sixty plus sixty equals zero. Negative one hundred eighty plus three hundred equals one hundred twenty. So we get eight x equals one hundred thirty five x plus one hundred twenty.
Step four is to move all x terms to one side. We subtract one hundred thirty five x from both sides. Eight x minus one hundred thirty five x equals negative one hundred twenty seven x. So we get negative one hundred twenty seven x equals one hundred twenty. Step five is to isolate x by dividing both sides by the coefficient of x, which is negative one hundred twenty seven. This gives us x equals one hundred twenty divided by negative one hundred twenty seven, or negative one hundred twenty over one hundred twenty seven.
To summarize our solution: We cleared the fractions by multiplying the entire equation by the least common multiple of sixty. We distributed the terms and combined like terms to simplify both sides. We isolated x by moving all variable terms to one side. Our final answer is x equals negative one hundred twenty over one hundred twenty seven.